KATHOLIEKE UNIVERSITEIT LEUVENFACULTEIT TOEGEPASTE WETENSCHAPPENDEPARTEMENT BURGERLIJKE BOUWKUNDELABORATORIUM REYNTJENSKasteelpark Arenberg 40, B - 3001 Heverlee (Belgium)

Probabilistic evaluation of structuralunreinforced masonry

Promotor: Proefschrift voorgedragen totprof. dr. ir. D. Van Gemert het behalen van het doctoraat

in de toegepaste wetenschappen

door

ir. Luc Schueremans

December 2001

KATHOLIEKE UNIVERSITEIT LEUVENFACULTEIT TOEGEPASTE WETENSCHAPPENDEPARTEMENT BURGERLIJKE BOUWKUNDELABORATORIUM REYNTJENSKasteelpark Arenberg 40, B - 3001 Heverlee (Belgium)

Probabilistic evaluation of structuralunreinforced masonry

Jury Members:

prof. dr. ir. E. Aernoudt, Chairman Dissertation submitted to theprof. dr. ir. D. Van Gemert, Promotor Faculty of Applied Scienceprof. dr. ir. G. De Roeck, Assessor for the Degree of Doctor in prof. dr. ir. J. Van Dyck, Assessor Civil Engineeringprof. dr. ir.-arch. K. Van Balen prof. dr. ir. M. Maes, University of Calgary, Canadaprof. dr. ir. L. Taerwe, R.U.Gent by

ir. Luc Schueremans

U.D.C. 624.012

December 2001

OnderzoeksinstellingKatholieke Universiteit LeuvenDepartement Burgerlijke BouwkundeAfdeling Bouwmaterialen en BouwtechniekenLaboratorium ReyntjensKasteelpark Arenberg 40B - 3001 Heverlee - België

Research InstituteKatholieke Universiteit LeuvenDepartment of Civil EngineeringDivision of Building Materials and Building TechnologyReyntjens LaboratoryKasteelpark Arenberg 40B - 3001 Heverlee - Belgium

Beursverlenende instantieInstituut voor de aanmoediging van Innovatie door Wetenschap en Technologie in Vlaanderen (IWT-Vlaanderen)Bischoffsheimlaan 25B - 1000 Brussel - België

Research grant offered byInstitute for the encouragement of Innovation by Scienceand Technology in Vlaanderen (IWT-Vlaanderen)Bischoffsheimlaan 25B - 1000 Brussels - Belgium

©2001 Faculteit Toegepaste Wetenschappen, Katholieke Universiteit Leuven

Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaargemaakt worden door middel van druk, fotokopie, microfilm, elektronisch of op welke anderewijze ook zonder voorafgaande schriftelijke toestemming van de uitgever, DepartementBurgerlijke Bouwkunde, Katholieke Universiteit Leuven, Kasteelpark Arenberg 40, 3001Heverlee, België.

All rights reserved. No part of this book may be reproduced, stored in a database or retrievalsystem or published in any form or in any way - electronically, mechanically, by print,photoprint, microfilm or by any others means - without the prior written permission of thepublisher, Department of Civil Engineering, Katholieke Universiteit Leuven, KasteelparkArenberg 40, 3001 Heverlee, Belgium.

D/2001/7515/35ISBN 90-5682-327-2

i

Voorwoord

Met dit doctoraat sluit ik een periode van 6 jaar onderzoek aan het Departement Bouwkunde af.Een gelegenheid om de vele mensen te bedanken voor hun ondersteuning in de totstandkomingvan dit werk.

Dionys Van Gemert heeft me de ruimte gelaten mijn eigen weg te zoeken in ditonderzoeksdomein, zonder evenwel het doel uit het oog te verliezen. Ook naast hetpromotorschap wil ik mijn waardering uitdrukken. De veelzijdige onderzoeksprojecten en“oranje mapjes” hebben gezorgd voor een boeiende verrijking als ingenieur, door de begeleidingvan projectonderwijs en oefenzittingen heb ik een band met het onderwijs kunnen behouden.

De assessoren Guido De Roeck en Jozef Van Dyck wil ik bedanken voor hun grondige lectuuralsook feedback bij voorliggende tekst. Hun kritische noten hebben mij verplicht het conceptvanuit meerdere invalshoeken te herdenken.

Bijzondere dank gaat ook uit naar Marc Maes. Mijn verblijf aan de University of Calgary,Canada, in 1998 heeft het onderzoek een belangrijke duw in de rug gegeven. Daarnaast heb ikmet volle teugen kunnen genieten van het prachtige land en zijn aangename inwoners.

Ook Paul Waarts en Ton Vrouwenvelder wil ik danken voor de mogelijkheid deeltijds onderzoekte verrichten aan TNO, Nederland. Het doctoraatswerk van Paul Waarts en de discussies overhet onderzoeksdomein vormden een springplank voor het eigen onderzoek.

Ik wil ook graag de leden van de jury bedanken: Koen Van Balen, Luc Taerwe en Marc Maes.Afgelopen 6 jaar heb ik de gelegenheid gehad om met hen meermaals van gedachten te wisselen,nieuwe ideeën op te doen en met steeds meer motivatie door te werken. Ik ben vereerd dat zijin mijn jury zetelen. Ik dank ook Etienne Aernoudt voor het waarnemen van de voorzittertaak.

De collega’s van de afdeling bouwmaterialen en bouwtechnologie en de mensen van hettechnisch personeel draag ik een warm hart toe. Aan de vele leuke momenten op het werk en deaangename samenwerking behoud ik mooie herinneringen. Zij hebben elk op hun manier eenwezenlijke bijdrage geleverd.

Tot slot wil ik een woord van dank richten tot vrienden en familieleden. Hoewel ze nietrechtstreeks bij dit werk betrokken waren, zijn ze van onschatbare waarde voor mij.

Ouders en schoonouders vernoem ik in één adem. Hun steun, enthousiasme en bezorgdheidlijken wel onuitputtelijk.

Een bijzondere plaats behoud ik voor Pascale en onze twee kleine spruiten, Judith en Bas. Zijzijn mijn zekerheden in dit onzekere bestaan.

Luc Schueremans, December 2001

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Nederlandstalige Abstract

Het doel van dit onderzoek is tweeledig. Enerzijds wordt een algemene methodologie opgesteldom objectieve veiligheidswaarden te berekenen voor bestaande structuren, anderzijds spitst hetonderzoek zich toe op ongewapende historische metselwerkstructuren.

Vooreerst worden mogelijke doelwaarden voor de veiligheid of betrouwbaarheid aangereikt. Omhet huidige veiligheidsniveau te berekenen, wordt een niveau III methode of probabilistischemethode vooropgesteld. Na een overzicht van de bestaande betrouwbaarheidstechnieken, komenrecent ontwikkelde technieken aan bod.

Het onderzoek beoogt de berekening van de veiligheid van gehele systemen. De kans dat degrenstoestandsfuncties van het systeem worden overschreden, wordt berekend. In eenvoudigegevallen zal deze grenstoestand in een analytische vorm beschikbaar zijn, in andere niet. In datlaatste geval zal een numerieke techniek, bijvoorbeeld een niet-lineaire eindige elementenberekening, vereist zijn. Deze berekeningen kunnen veel tijd vergen en daarom is het nuttig omover een techniek te beschikken die het aantal oproepen naar deze grenstoestandsfunctiesminimaliseert. De grenstoestanden voor metselwerk worden gebundeld en aangepast waar nodigom gehanteerd te worden in een probabilistische procedure.

De recent ontwikkelde betrouwbaarheidstechnieken hebben tot doel een systeembetrouwbaarheidte berekenen, met een minimum aantal expliciete oproepen naar de grenstoestandsfunctie. Zijbestaan veelal uit een combinatie van de traditionele methodes, aangevuld met het gebruik vaneen aanpasbaar responsoppervlak. Deze methodes worden verder uitgewerkt en verbeterd.

In de niveau III methode kunnen alle probleemvariabelen gedefinieerd worden alstoevalsvariabelen om de aanwezige onzekerheid in rekening te brengen. Het betreft:- de belastingen (krachten en verplaatsingen zoals zettingen),- de geometrie (dwarsdoorsnede, excentriciteiten),- de materiaaleigenschappen zoals sterkte- en stijfheidseigenschappen.

De materiaaleigenschappen komen uitgebreid aan bod in het experimenteel luik. Dewaarschijnlijkheidsverdelingen van de belangrijkste materiaaleigenschappen worden opgesteld.Niet enkel metselwerk in druk, maar ook het gedrag in trek, dwarskracht en multi-axialespanningstoestand komen aan bod. Voor deze laatste wordt gebruik gemaakt van een triaxiaalceldie speciaal voor het onderzoek van heterogene materialen werd aangeschaft.

De techniek wordt toegepast op verscheidene voorbeelden, waaronder de veiligheid vanmeerschalig metselwerk, een excentrisch belaste kolom, bogen en afschuifwanden. Hiermeekomen zowel analytische als numerieke problemen aan bod. Met de veiligheid vanafschuifwanden wordt een link gelegd naar meer hedendaags metselwerk.

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English Abstract

The scope of this thesis is twofold. On the one hand, a general methodology is looked for toobtain objective safety values for existing structures, on the other hand, the methodology focuseson unreinforced historical masonry structures.

Possible target values for the failure probability or reliability index are given. To calculate thepresent safety level, a level III method or probabilistic method is proposed. Traditional reliabilitymethods are reviewed and recent developments are treated.

The safety value, failure probability or reliability index of a whole system or structure or astructural element is aimed at. Calculating the failure probability means calculating theprobability that a limit state function is exceeded. In some cases the limit state function is knownanalytically, in other cases each limit state function evaluation will require a (non-linear) finiteelement calculation. The latter may require a substantial amount of calculation time. Therefor,a methodology that minimizes the number of direct limit state function evaluations is veryfruitful. The limit state functions for masonry and calculation models are retrieved and rephrasedfor assessment purposes where necessary.

Recently developed reliability methods aim at calculating a system reliability index, minimizingthe number of direct limit state function evaluations. This can be achieved by combiningtraditional reliability methods with an adaptive response surface. The new methodology isfurther developed and improved.

In case of a level III method, all variables are treated as random variables, accounting for thepresent uncertainties. These include:- loads (forces and displacements such as settlements),- geometrical properties (cross-sections, eccentricities)- material properties (strength and stiffness properties).

The material properties that govern the masonry material behavior are dealt with in theexperimental part. The probability distribution function of the main material properties aregathered. Not only masonry in compression is treated, but also masonry in tension regime, shearand multi-axial stress state. For the latter, a triaxial cell testing device was acquired capable ofhandling low strength heterogeneous materials.

The technique is illustrated on several examples: safety of three-leaf masonry, eccentricallyloaded masonry column, arches and shear walls. These cover analytical as well as numericalproblems. With the safety of shear walls, a link to more contemporary masonry is made.

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Table of ContentsVoorwoord - Foreword i

Nederlandstalige abstract - Dutch abstract iii

English abstract v

Table of contents vii

List of notations and abbreviations xiii

Nederlandstalige samenvatting - Dutch Summary xxi1. Inleiding xxi2. Betrouwbaarheidsmethodes - literatuuroverzicht xxiii

2.1. Algemeen betrouwbaarheidsprobleem, technieken en vereiste xxiiinauwkeurigheid

2.2. Analytische en numerieke integratie xxiv2.3. Monte Carlo xxiv2.4. Directional Integration en Directional Sampling xxv2.5. Eerste orde en tweede orde betrouwbaarheidsmethodes xxv

3. Betrouwbaarheidsmethodes die gebruik maken van een aanpasbaar xxvresponsoppervlak3.1. DARS xxvi3.2. Monte Carlo en FORM met een aanpasbaar responsoppervlak (ARS) xxvii3.3. Vergelijking met niveau I methodes xxix

4. Rekenmodellen voor metselwerk xxix5. Experimenteel onderzoek xxxi

5.1. Componenten baksteen en mortel xxxii5.2. Composiet metselwerk xxxiii5.3. Metselwerk in druk - stochastische uitbreiding xxxvi5.4. Homogenisering - stochastische uitbreiding xxxvii5.5. Metselwerk in afschuiving xxxviii5.6. Metselwerk in trek xxxix5.7. Metselwerk in een multi-axiale spanningstoestand xl

6. Toepassingen xlii6.1. Consolidatie van meerschalig metselwerk xlii6.2. Een kolom belast op een excentrisch aangrijpende drukkracht xliv6.3. Metselwerk bogen xlv6.4. Metselwerk afschuifwanden xlvii

7. Besluiten en verder onderzoek l7.1. Besluiten l7.2. Verder onderzoek l

8. Appendix A li9. Appendix B lii

1. Introduction 1

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1.1. General framework 11.1.1. The need for assessment 11.1.2. The objectives of a reliability assessment 21.1.3. The assessment framework 31.1.4. Different levels to assess a structures’ safety 31.1.5. Target safety for existing (historical) buildings 5

1.1.5.1. Target failure probabilities 61.1.5.2. Target reliability index 8

1.2. Building Materials and Building Technology Division 91.3. Scope of this thesis 101.4. Outline of this thesis 11

2. Reliability analysis - literature review 132.1. Introduction 132.2. Formulation of the generalized reliability problem 132.3. Required accuracy 162.4. Overview of standard reliability methods 172.5. Analytical and numerical integration 202.6. (Importance Sampling) Monte Carlo 22

2.6.1. Crude Monte Carlo 222.6.2. Importance Sampling Monte Carlo 23

2.7. Directional integration and Directional (Importance Sampling) 272.7.1. Directional Integration 272.7.2. Directional Sampling 282.7.3. Importance Sampling 29

2.8. First order and second order reliability methods 312.8.1. FOSM - First Order Second Moment method 312.8.2. FORM - First Order Reliability Method 372.8.3. SORM - Second Order Reliability Method 382.8.4. System analysis (SA) 41

2.9. Conclusions 44

3. Reliability analysis using an adaptive response surface 473.1. Introduction 473.2. Implicit limit states - Response surface methodology 47

3.2.1. Selection of random variables 483.2.2. Response Surface - functional form 503.2.3. Design of Experiments 513.2.4. Validation of the Response Surface 523.2.5. Design updating 53

3.3. DARS - Directional Adaptive Response Surface Sampling 543.3.1. DARS - Governing relations 553.3.2. DARS - Validation 593.3.3. DARS - Remarks 613.3.4. DARS (λadd=var) 62

3.4. MC or FORM and an Adaptive Response Surface (ARS) 663.4.1. Monte Carlo and an Adaptive Response Surface (MCARS) 66

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3.4.2. FORM and an Adaptive Response Surface (FORMARS) 723.5. Comparison with level I methods 733.6. Conclusions 74

4. Structural modeling of masonry 774.1. Introduction 774.2. Masonry subjected to a vertical loading 78

4.2.1. Analytical models 794.2.2. Vertical loading - Design model according to EC6 84

4.3. Masonry arches 854.3.1. Thrust lines 854.3.2. Arching - Design model according to EC6 87

4.4. Lateral loading (wind) and shear walls 884.4.1. Unreinforced walls subjected to lateral loads - design model according

to EC6 894.4.2. Unreinforced masonry shear walls - design model according to EC6 91

4.5. Finite element modeling of masonry 924.5.1. Macro-model - anisotropic continuum modeling 934.5.2. Masonry in compression 954.5.3. Masonry in tension 964.5.4. Masonry in shear 97

4.6. Material properties 974.7. Conclusions 98

5. Experimental research 995.1. Introduction 995.2. Components - brick and mortar 103

5.2.1. Bricks 1035.2.2. Cores ‰50mm with a height of 44 mm 1045.2.3. Couplets with a height of 120 mm 1075.2.4. Prisms sawn from bricks with a height of 160 mm 1095.2.5. Discussion of results - compressive strength fc 1095.2.6. Mortar 112

5.3. The composite masonry - masonry in compression 1145.3.1. Small masonry pillars 1145.3.2. Masonry cores 1165.3.3. Masonry Wallets 1175.3.4. Influence of test bank and measurement setup 1195.3.5. Effect of sample size and boundary conditions on the compressive

strength fc 1205.4. Compressive strength of masonry - stochastic extension 1235.5. Homogenization - stochastic extension 126

5.5.1. Homogenization - State of the art 1265.5.2. Governing relations - 2D elastic formulation 1275.5.3. 2D - Stochastic extension 131

5.6. Masonry in shear regime 1335.6.1. Masonry cores 135

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5.6.2. Wallets 1375.7. Masonry in tension 1395.8. Masonry in a multi-axial stress state 142

5.8.1. Introduction 1425.8.2. Triaxial cell testing device - test results 1435.8.3. Test results - mortar 1455.8.4. Test results - masonry 148

5.9. Conclusions 154

6. Applications 1576.1. Introduction 1576.2. Consolidation of three-leaf masonry walls - grout injection 157

6.2.1. Problem definition 1576.2.2. Material model 1586.2.3. Reliability analysis 1606.2.4. Conclusions 163

6.3. Masonry column subjected to an eccentric vertical load 1636.3.1. Problem definition 1636.3.2. Ultimate Limit State - probabilistic evaluation on level III 1656.3.3. Ultimate Limit State - checkpoints according to EC1 - level I 1686.3.4. Serviceability Limit State - probabilistic evaluation on Level III 1696.3.5. Time dependency 1716.3.6. Conclusions 172

6.4. Safety of masonry arches 1736.4.1. Problem definition 1736.4.2. Evaluating the safety of an existing arch 1756.4.3. Reliability analysis - DARS and MCARS+VI 1766.4.4. Upgrading the reliability index - different possibilities 1816.4.5. Conclusions 184

6.5. Masonry Shear Walls 1856.5.1. Problem definition 1856.5.2. Finite element modeling 1876.5.3. Reliability analysis -ULS 1886.5.4. Reliability analysis - SLS 1926.5.5. Reliability analysis - analytical model according to EC6 1936.5.6. Conclusions 194

6.6. Conclusions 195

7. Conclusions and further research 1977.1. General conclusions 1977.2. Further research 201

8. References 203Annex A - Academic examples 225

A.1. Standard R-S problem 227A.2. Noisy limit state function 228A.3. R-S problem with quadratic term 229

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A.4. Limit state function with 10 quadratic terms 230A.5. Limit state function with 25 quadratic terms 231A.6. Convex failure domain 232A.7. Oblate Spheroid 233A.8. Saddle surface 234A.9. Discontinuous limit state function 235A.10. Two branches 237A.11. Concave failure domain 238A.12. Series system with 4 branches 239A.13. Parallel system 240A.14. Altered series system with 4 branches - described in Chapter 2 and Chapter 3241A.15. Reliability of a plane frame - DARS-CALFEM: a stochastic finite element 242

method

Annex B - Experimental results - summary and statistical processing 245B.1. Cores ‰50 with a height of 44 mm 246B.2. Couplets with a height of 120 mm 247B.3. Prisms sawn from bricks with a height of 160 mm 249B.4. Mortar samples - l×w×h = 160×40×40 mm3 251B.5. Masonry pillars 254B.6. Masonry cores ‰=150 mm - vertically drilled 256B.7. Masonry wallets - uni-axial compressive test 256B.8. Masonry cores ‰=150 mm - diagonally drilled 257B.9. Masonry wallets - shear test 257B.10. Masonry cores - tension 259B.11. Triaxial testing - mortar cores 261B.12. Triaxial testing - vertically drilled masonry cores 262B.13. Triaxial testing - diagonally drilled masonry cores 263

Curriculum Vitae 265

List of Publications 267

Acknowledgment 273

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List of notations and abbreviations

Lower cases

a,b numerical parameterai degradation rate for parameter iarot distance chosen by the rotatability requirementc cohesioncov coefficient of variationd thicknessdc crack depthdel depth of elastic areadef effective thicknessdk search direction in kth iterationdi thickness of layer idpl depth of plastic areadr 2nd order deviation with respect to perfect circular form in arch analysise eccentricityei auxiliary strainsfX(x) marginal probability distribution function of random variable X fb brick compressive strengthfb,m brick mean compressive strengthfc compressive strengthfext compressive strength of the external leaf of a multi-leaf wallfft flexural strengthfgr compressive strength of the injection groutfinf,0 compressive strength of the infill material of a multi-leaf wall before

strengtheningfinf,inj compressive strength of the injected infill materialfinf,s compressive strength of the infill material of a multi-leaf wall after

strengtheningfk characteristic strengthfm mortar compressive strengthfm,45° mortar compressive strength under 45° with respect to material axisft tensile strengthfv shear strengthfwc,0 compressive strength of multi-leaf wall before strengtheningfwc,s compressive strength of multi-leaf wall after strengtheningfx flexural strengthg limit state functiongULS Ultimate Limit State functiongSLS Serviceability Limit State functionh heighthef effective heighthv(v) importance sampling function

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l lengthlc length of area in compressionlref reference lengthm(u) merit functionn 1number of random variables,

²number of test samplesnLSFE number of limit state function evaluationsnp number of personsnroot number of roots found during the sampling proceduren2 number of central points in optimal design of experimentsnβ number of samples required to limit a type II error to a level βpf failure probability

sample failure probability�pf

pS safety probabilitypf,N nominal failure probabilitypf,T target failure probabilityp-value probability on a type I errorr radiusr0 radius of archrα bending moment coefficientrg shear retention factorrµ orthogonal strength ratio of the characteristic flexural strengths of masonrys sample standard deviationsl step length sl=[0...1]t timeth threshold, number of extremes used in UH-plotti auxiliary stressestL (design) service lifet0 quantile for test statistic evaluated for null hypothesis tα quantile with significance level α u vector of random variables in u-space (standard normal space)u’ vector of random variables in u’-space (rotated standard normal space)u, ui random variables in u-space (standard normal space)u* design point, most likely failure point in u-spacev velocityw widthwc crack widthx vector of values of random variables X in x-spacex, xi possible values of random variable X in x-spacex* design point, most likely failure point in x-space

expansion point for Taylor series expansion~xxc centre point for design of experiments

sample meanxy yearszβ z-value (standard normal distribution) for a significance level β

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Uppercases

A,B,C numerical parameterA Hessian matrix (with normalized second order derivatives)A observation matrixA surface, cross sectionAc activity factorC stiffness matrixCf cost factorCh homogenized stiffness matrixD diameterE Young’s modulus Ex, Ey according to x,y-material axisEy,LVDT Young’s modulus in y-axis according to plate displacementE[ ] expected valueF F-distribution typeFα α=upper quantile of F-distributionFc compressive forceFt tensile forceFV vertical forceFX(x) cumulative distribution function of random variable XG shear modulusGfc fracture energy in compressive regimeGf,I mode I (tension) fracture energyGf,II mode II (shear) fracture energyH0 null-hypothesisH1 alternative hypothesisI moment of inertiaI [ ] indicator functionK factor accounting for brick type diversification in masonry strength calculationM momentum forceN 1number of random samples,

²normal probability distribution functionNRd normal Design Resistance force of a wallPe strain Projection matrixPf failure probabilityPt stress Projection matrixR global symbol for a random resistance variable referring to the standard R-S

problemR orthonormal transformation matrixRd design resistance variableS 1global symbol for a random load variable referring to the standard R-S

problem²estimator of standard deviation

S2 estimator of varianceSc social criterionSp pooled estimator of standard deviation

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T test statistic of Student distributionV total volume of a multi-leaf wallV() coefficient of variation of () V0 initial volume of voids in the inner leaf of loose rubble material Vext volume of the external leafs of a multi-leaf masonry wallVinf volume of the infill material, inner leaf of loose rubble materialVinj injected volumeW warning factorWk characteristic wind load per unit areaX random variable in original spaceX vector of random variables in the original spaceY outcome, based on estimated response surface~LN lognormal distributed random variable~N normal distributed random variable x~N(µ(x),σ(x)²)

Greek - lower cases

α 1confidence, significance level for hypothesis tests of confidence intervals²angle

αg geometrical safety factorai

1direction cosine, with respect to the variable i²importance factor when used as checkpoints according to EC1 (EC1,1994)

αS static safety factorβ 1reliability index

²Significance level in hypothesis test for type II errorβa actual reliability indexβFOSM First Order Second Moment reliability indexβFORM First Order Reliability Method reliability indexβSORM Second Order Reliability Method reliability index βRS,SORM reliability index based on the Response Surface (RS), calculated using a

Second Order Reliability Method (SORM)βT target reliability indexδ normalizing factor for bricks according to EC6∆g,add additional distance in MCARS+VI reliability procedureζ parameter of Generalized Pareto Distribution (GDP)η parameter of Generalized Pareto Distribution (GDP)ε 1error term, model uncertainty, normal distributed random variable ~N(µε,σε²)

²Strainεg,i error between outcome RS and real LSF for sample iεh homogenized strainsεL lack of fit error termεnn vertical dilatationεr residual error termεv vertical strains

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ε0 pure error termφ friction angleφ marginal normal distribution functionγ shear angle γm material partial safety factorγF load partial safety factorθθθθ (polar coordinates) unit vector which defines a direction in the standard

normal spaceθi regression coefficient iλ 1length of vector in the standard normal space, more specific used as length to

the root of the limit state function in the standard normal space in case ofDirectional Sampling procedures²slenderness

λadd additional distance in the u-space used in DARS to distinguish the importantfrom the non-important region

λadd,fin the final additional distance in the u-space used in DARS to distinguish theimportant from the non-important region, obtained at the end of the iterationprocedure

λADI,min minimum distance in the u-space found during the Axis DirectionalIntegration procedure (ADI)

λeq equivalent slendernessλmin minimum distance in the u-space found during the sampling procedure (DS or

DARS)µ mean valueν Poisson’s ratioρ 1correlation,

²densityρn reduction factor that accounts for the stiffening of a wall at the loaded endsσ 1standard deviation

²stressσc confining pressureσh standard deviation of the sampling function (hv(v)~N(µh,σh²)), used in Variance

Increase (VI)σh homogenized stressesσh,fin final standard deviation of the sampling function (hv(v)~N(µh,σh²)), used in

Variance Increase (VI) obtained at the end of the sampling procedureσv vertical stressτ shear stressχ²n Chi-square distribution type with n degrees of freedomκi curvature i

Greek - upper cases

Φ cumulative normal distribution

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Φi,m capacity reduction factor

Scientific

‰ diameterj Partial derivativesi Jacobian, matrix with first order partial derivativesi² matrix with second order partial derivatives

Abbreviations

ADI Axis Directional IntegrationANOVA Analysis of VarianceARS Adaptive Response Surface CCD Central Composite DesignCI Confidence IntervalCPU Computer Processing UnitsDARS Directional Adaptive Response surface SamplingDS Directional SamplingEV-I Extreme value distributions of type I (Gumbel), such as: exponential, gamma,

Weibull, normal, lognormal, logisticEV-II Extreme value distributions of type II (Frechet), such as: Pareto, Cauchy, log-

gammaFORM First Order Reliability MethodGPD Generalized Pareto Distribution type (extreme value distribution type)GUI Graphical User InterfaceISMC Importance Sampling Monte CarloLA Limit Analysis used for the calculation of thrust lines in masonry archesLN Lognormal distribution typeLSF Limit State FunctionLSFE Limit State Function EvaluationLVDT Linear Varying Deformation TransducerMC Monte CarloMCARS+VI Monte Carlo Adaptive Response surface Sampling with Variance IncreaseMSE Mean Squares (pure error)MSL Mean Squares (lack of fit)MSR Mean Squares (total error)MVFM Mean Value First Moment reliability indexNAD National Application DocumentNTM Non Tension MaterialPDF Probability Distribution FunctionQQ-plot Quantile-Quantile plotRAM Random Accessible MemoryRH Relative Humidity

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RS Response SurfaceRVE Representative Volume ElementSLS Serviceability Limit StateSSE Sum of Squares (lack of fit)SSL Sum of Squares (pure error)SSR Sum of Squares (total error)SORM Second Order Reliability MethodUH-plot gereralized quantile plotULS Ultimate Limit State

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Nederlandstalige samenvatting

1. Inleiding

Instortingen zoals die van de gewelven van de Sint-Fransiscus basiliek (Italië, 1997), de toren vanPavia (Italië, 1989) of de toren van Chichester Cathedral (Engeland, 1861), bewijzen datveiligheid, betrouwbaarheid en risico sleutelbegrippen zijn in het behoud van het bouwkundigerfgoed. Het bouwkundig erfgoed in België is uitzonderlijk rijk. Met ongeveer 10 000beschermde monumenten vraagt het behoud hiervan belangrijke inspanningen. De KoningBoudewijn Stichting rekent voor dat een bedrag van 2.25 miljard Euro nodig is om debouwkundige staat van het beschermde monumentenbestand zo te verbeteren, dat bij normaalonderhoud de eerste vijftig jaar geen ingrijpende restauratiewerken nodig zijn.

Veiligheid, zowel in de uiterste grenstoestand als gebruiksgrenstoestand, maakt dan ook nietlanger enkel deel uit van het ontwerp. Steeds meer wordt aandacht besteed aan de evaluatie vanbestaande structuren. Deze evaluatie kan gevraagd worden omwille van een schare vanoorzaken, waaronder: een verandering in gebruik, het overschrijden van de vooropgesteldelevensduur, een controle na een specifieke ramp (aardbeving) of vastgestelde schade.

In het conservatieproces komt de betrouwbaarheidsanalyse meermaals voor. De aandacht vandit onderzoek gaat in de eerste plaats naar de gedetailleerde structurele analyse van de bestaandetoestand, waarbij een objectief antwoord wordt gezocht op de vraag naar het huidigeveiligheidsniveau. Daarnaast heeft de betrouwbaarheidsanalyse belangrijke troeven bij deverdere stappen in het conservatieproces. Zowel bij het voorstellen van de therapie als decontrole, is het interessant te weten in welke mate het veiligheidsniveau wordt beïnvloed dooreen ingreep.

Doorheen de geschiedenis werden structuren ontworpen overeenkomstig de dan geldende normenen beschikbare kennis. Eurocode 1 ordent deze evolutie in ontwerpregels van niveau 0 tot niveauIV. De niveau III en IV methodes zijn de meest objectieve en nauwkeurige. Ze berekenen deexacte faalkans van een geheel structureel systeem of structureel element, gebruik makend vande exacte waarschijnlijkheidsfunctie van alle toevalsvariabelen. Het is dan ook dezeprobabilistische procedure die wordt nagestreefd in deze studie. Niveau IV methodes voegenkosten-baten elementen toe aan de probabilistische procedure. Dit valt buiten het doel van ditonderzoek.

Voor de beoogde veiligheid worden een aantal streefwaarden vooropgesteld. Uitgaande vanliteratuurvoorbeelden wordt een globaal model aangereikt dat rekening houdt met uiteenlopendeinvloedsfactoren, zoals: type van mogelijke schade, voorziene levensduur van de structuur, hetaantal mensen dat blootgesteld is aan risico, de mate waarin mensen worden gewaarschuwd voornakend gevaar. Om de waarde van het gebouw in het kader van het erfgoed te plaatsen, wordtéén van de evaluatiefactoren geherdefinieerd (Sc). Streefwaarden voor globaleveiligheidsniveau’s kunnen bepaald worden volgens Vgl. 1 en Tabel 1:

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(1)p S tn

AW

CfT cL

p

cf= −10 4

Kostenfactor Cf Waarschuwingsfactor Wniet ernstigernstigzeer ernstig

1010.1

- gefaald-veilig conditie- gradueel falen met zekere waarschuwing- gradueel falen verborgen aan het zicht- plots falen, zonder waarschuwing

0.010.10.31.0

Activiteitsfactor Ac Sociaal criterium Sc

- Post-ramp activiteit- Normale activititeiten: Gebouwen Bruggen- Structuren met hogeblootstelling (off-shore)

0.3

1.03.010.0

- publieke verzamelplaatsen, stuwdam,(historische gebouwen, met groot belang voorde mensheid of opgenomen in de UNESCOlijst bv.)- woningen, kantoren, handelspanden(geschermde monumenten)- bruggen- torens, masten, off-shore constructies

0.005

0.05

0.55

Tabel 1: Invloedsfactoren bij het bepalen van nominale streefwaarden voor de faalkans

Huidige normen die verwijzen naar niveau II en III berekeningen, hanteren over het algemeenbetrouwbaarheidsindices in plaats van faalkansen. Beide zijn gerelateerd via de standaardnormaal verdeling, Vgl. 2 en Tabel 2:

(2)( )pf = −Φ β

pf 10-1 10-2 10-3 10-4 10-5 10-6 10-7

β 1.3 2.3 3.1 3.7 4.2 4.7 5.2Tabel 2: Verband tussen β en pf

Het doel van dit onderzoek is tweeledig. Vooreerst wordt een algemene methodologie opgesteldom objectieve veiligheidswaarden te berekenen voor bestaande structuren. Vervolgens legt hetwerk zich voornamelijk toe op ongewapend historisch metselwerk. Deze algemene methodemoet toelaten de systeembetrouwbaarheid te berekenen van de ganse structuur of het structureelelement dat voorwerp is van de studie.

Het berekenen van de faalkans vereist de evaluatie van de grenstoestanden. In een aantalgevallen zijn deze grenstoestanden analytisch beschikbaar, in andere gevallen leidt dit tot (niet-lineaire) eindige elementen berekeningen. Vooral dan is het interessant over een methode tebeschikken die een minimum aantal rechtstreeks te berekenen grenstoestandsevaluaties vereist.

Om een objectieve veiligheidswaarde te berekenen, overeenkomstig niveau III, moeten allevariabelen als toevalsvariabelen kunnen ingevoerd worden. In het experimenteel onderzoek gaatde aandacht voornamelijk naar de materiaaleigenschappen.

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De methode wordt toegepast op uiteenlopende voorbeelden om zowel de kracht van demethodiek als de praktische toepasbaarheid te illustreren.

2. Betrouwbaarheidsmethodes - literatuuroverzicht

Dit hoofdstuk behandelt, samen met het volgende hoofdstuk, het berekenen van debetrouwbaarheid van structurele systemen. De bestaande methodes worden overlopen, de voor-en nadelen worden aangehaald en de methode wordt geïllustreerd aan de hand van eenacademisch voorbeeld. Verscheidene van deze methodes worden gebruikt in de toepassingen,Hoofdstuk 6 en Annex A.

2.1. Algemeen betrouwbaarheidsprobleem, technieken en vereiste nauwkeurigheid Voor het meest algemene geval is de faalkans gedefinieerd overeenkomstig:

(3)( )[ ] ( )( )

p P g f dfg

= ≤ =≤

X x xXX

00

...

waar g(X) de grenstoestandsfunctie aangeeft en fX(x) de gezamenlijkewaarschijnlijkheidsverdeling van de toevalsvariabelen X. Om de faalkans (pf) te berekenen, zijnde afgelopen 20 jaar verschillende standaardtechnieken ontwikkeld. Samen met de recenteontwikkelingen, Hoofdstuk 3, zijn deze aangegeven in Tabel 3.

Betrouwbaarheidsmethode - Niveau (I, II, III) - Direct/Indirect (D/ID)Integratiemethodes

Analytische en Numerieke Integratie (AI/NI, III, D)

Directional Integration (DI, III, D)

Samplingmethodes

(Importance Sampling) Monte Carlo ((IS)MC, III, D)

(Importance) Directional Sampling ((I)DS, II, D)

FORM/SORMmethodes

Eerste orde tweede momenten methode (FOSM, II, D)

Eerste orde en tweede orde betrouwbaarheidsmethodes (FORM/SORM)(niveau II) in combinatie met een systeemanalyse (FORM/SORM-SA, III, D)

Methodesmet eenaanpasbaarrespons-oppervlak

FORM/SORM met een aanpasbaar responsoppervlak (ARS) (niveau II) incombinatie met een systeemanalyse (III)(D-ID)

Directional Sampling met aanpasbaar responsoppervlak (DARS, III, D-ID)

Monte Carlo Sampling met aanpasbaar responsoppervlak (MCARS, III, D-ID)

Tabel 3: Overzicht van verschillende betrouwbaarheidsmethodes overeenkomstig niveau III

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In Hoofdstuk 2 en 3 worden de methodes geïllustreerd op een academisch voorbeeld, Vgl. 4. Ditis reeds behandeld door verscheidene auteurs. Het is voldoende eenvoudig om illustratief teblijven en bevat de nodige complexiteit om de kracht of zwakte van de verschillende methodesaan te geven (u1~N(0,1) en u2~N(0,1)):

(4)( )

( ) ( ) ( )

( ) ( ) ( )

( )( )

g u u

g u u u uu u

g u u u uu u

g u u u ug u u u u

1 2

1 1 2 1 22 1 2

2 1 2 1 22 1 2

3 1 2 1 2

4 1 2 2 1

2 0 012

2 0 012

2 5 22 5 2

, min

, . .

, . .

, ., .

=

= + − −+

= + − ++

= − += − +�

Om de verschillende methodes objectief met mekaar te kunnen vergelijken, wordt een criteriumvoorgesteld dat de vereiste nauwkeurigheid van de resulterende betrouwbaarheidsindex vastlegt:

(5)( ) ( )σ β β β β= ≤ = ≥015 3 00 0 05 3 00. , . : . , .voor en V voor

2.2. Analytische en Numerieke IntegratieAnalytische en Numerieke Integratie (AN/NI) zijn beide niveau III methodes. Het belangrijkstenadeel is het aantal grenstoestandsevaluaties dat vereist is met betrekking tot het aantaltoevalsvariabelen in het probleem (n): 9n, voor β=4 en V(β)=0.05. Daarom worden zevoornamelijk gebruikt bij de validatie van andere methodes en bij problemen met een beperktaantal toevalsvariabelen (n@5).

2.3. Monte CarloDe elementaire Monte Carlo (MC) techniek is een niveau III methode. Het voornaamstevoordeel is de eenvoud van de techniek. Dat maakt de techniek zeer aantrekkelijk om tecombineren met eindige elementen methodes of een responsoppervlak. Het grootste nadeel ligtin het grote aantal simulaties dat vereist is om de vooropgestelde nauwkeurigheid te bereiken(NA3/pf). Tegemoetkomen aan dit nadeel kan gedeeltelijk door gebruik te maken van ImportanceSampling, waar de simulaties zich beperken tot het interessante gebied. Twee grote families vansimulatiefuncties kunnen worden onderscheiden. In de eerste wordt het gebied waar desimulaties worden uitgevoerd, verlegd naar de meest kritische zone (ISMC), waar het meestwaarschijnlijke faalpunt (x*) gelegen is. Deze techniek is interessant wanneer dit punt gekendis en wanneer dit het enige faalpunt is. In andere gevallen kan ze leiden tot inefficiëntie of zelfseen foutief resultaat. Globaal gezien blijft het aantal simulaties evenredig met het aantaltoevalsvariabelen: N=300n. In de tweede techniek wordt het domein waarin de simulatiesworden uitgevoerd, verbreed zodat men een grotere kans heeft om voldoende simulaties in hetfaaldomein te realiseren. Voor deze variantietoename (MC+VI) wordt een experimenteelbepaalde simulatiefunctie gebruikt:

(6)σ βh n= −0 4.

Het is duidelijk dat in beide gevallen enige voorkennis van het probleem vereist is.

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2.4. Directional Integration en Directional SamplingHet gebruik van poolcoördinaten leidt tot een serie technieken die gelijkaardig zijn aanbovengenoemde. Directional Integration (DI) en Directional Sampling (DS) hebben dan ookvergelijkbare voor- en nadelen en een vergelijkbare efficiëntie. Directional Sampling heeft hetbijkomend voordeel dat de nulpunten van de grenstoestanden berekend worden. Dit is de grenstussen het veilige en onveilige gebied. Uiteraard is dit met betrekking tot structureel inzichtbelangrijke informatie.

Gemiddeld is voor het bepalen van dit nulpunt volgens een lukraak gekozen richting een drietalgrenstoestandsevaluaties vereist. Omdat het aantal grenstoestandsevaluaties proportioneel is methet aantal toevalsvariabelen, blijft deze techniek interessant voor problemen met een groter aantaltoevalsvariabelen. Gezien de efficiëntie van de techniek, komt ze zeker in aanmerking omgecombineerd te worden met een responsoppervlak. Voor β=4 bedraagt het gemiddeld aantalsimulaties ongeveer N=160n wat resulteert in 480n grenstoestandsevaluaties (LSFE).

2.5. Eerste orde en tweede orde betrouwbaarheidsmethodesEerste orde en tweede orde betrouwbaarheidsmethodes (FOSM/FORM/SORM) zijn niveau IImethodes. In plaats van benaderende methodes te gebruiken om de uitkomst van de faalkans tebegroten, wordt bij deze technieken de integrand in het faalpunt vereenvoudigd tot een eerste oftweede orde veelterm.

De eerste orde tweede momenten methode (FOSM) beperkt zich tot de eerste twee momenten,gemiddelde en standaard deviatie en is daardoor slechts benaderende voor toevalsvariabelen dieniet normaal of Gaussiaans verdeeld zijn. Met de eerste orde en tweede orde methodes(FORM/SORM) wordt aan dit euvel verholpen door het invoeren van een normaal-staarttransformatie. Daarbij wordt de eerste orde of tweede orde veeltermbenadering gecentreerd rondhet faalpunt (x*), wat de methode invariant maakt voor de gekozen probleemdefinitie.

Hoewel het zeer efficiënte methodes betreft met betrekking tot het aantal rechtstreeksegrenstoestandsevaluaties, is een systeemanalyse vereist om een niveau III schatting te verkrijgenvan de faalkans. Omdat enkel een boven- en ondergrens berekend wordt, beperkt dit het gebruikin combinatie met andere methodes, ondanks hun intrinsieke efficiëntie. Deze efficiëntie wordtgeïllustreerd op 14 voorbeelden in Annex A.

3. Betrouwbaarheidsmethodes die gebruik maken van eenaanpasbaar responsoppervlak

Dit hoofdstuk behandelt recente ontwikkelingen in betrouwbaarheidsmethodes. De klassiekemethodes, beschreven in Hoofstuk 2, worden gecombineerd met een aanpasbaarresponsoppervlak (ARS).

Extreem geïdealiseerde systemen kunnen op een analytische wijze bestudeerd worden, zoweldeterministisch als probabilistisch. Een reëel systeem heeft echter een hoge graad vancomplexiteit. Een mogelijke oplossing voor deze complexe systemen ligt, zo ook voor eendeterministische analyse, bij de numerieke methodes, waaronder eindige elementen methodes.

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Bij het gebruik van een responsoppervlak wordt de respons, die een complexe functie is van deinputvariabelen, benaderd door een eenvoudig verband van deze inputvariabelen. Wanneer ditresponsoppervlak gerealiseerd is, dan kan de betrouwbaarheidsanalyse worden uitgevoerd op hetresponsoppervlak, in plaats van op het originele probleem. Gezien het responsoppervlak eeneenvoudig functioneel verband is, vereist dit slechts een fractie van de rekentijd.

3.1. DARSHoewel het gebruik van een responsoppervlak veelbelovend is, heeft deze techniek een grootnadeel. Het aantal directe grenstoestandsevaluaties nodig om een accuraat responsoppervlak tefitten, is evenredig met ~2n. Daardoor dreigt de techniek enkel een verschuiving vangrenstoestandsevaluaties mee te brengen. Dit zou deze methode weinig aantrekkelijk makenvoor toepassingen met een groot aantal toevalsvariabelen.

Om het aantal grenstoestanden tot een minimum te beperken, wordt de techniek gecombineerdmet een van de traditionele betrouwbaarheidsmethodes. Waarts was de eerste om DirectionalSampling te combineren met een aanpasbaar responsoppervlak (DARS). Globaal bestaat detechniek uit 3 stappen:• Stap 1: naast een grenstoestandsevaluatie in de oorsprong van de standaard normaal

ruimte (u-ruimte), wordt de waarde van elke toevalsvariabele individueel verhoogd, tothet nulpunt (λ) van de grenstoestand (grens tussen veilig en onveilig) is teruggevonden.Dit kan gezien worden als een vorm van Directional Integration volgens de hoofdassen(ADI).

• Stap 2: een eerste responsoppervlak wordt gefit doorheen deze data op basis van eenkleinste kwadraten benadering.

• Stap 3: deze stap is een iteratieve stap. Het responsoppervlak wordt aangepast en defaalkans of betrouwbaarheidsindex worden geupdated tot de vereiste nauwkeurigheid isbereikt. Daarvoor wordt de elementaire Directional Sampling toegepast op hetresponsoppervlak. Voor elke simulatie wordt eerst het nulpunt op het responsoppervlakbepaald. Omdat dit een lage orde veelterm is, is de rekentijd minimaal. Vervolgenswordt gecontroleerd of de desbetreffende richting een belangrijke richting is: λRS<>λmin+λadd. Waarts stelt voor om λadd=3.0 te nemen. Is de afstand kleiner, dan betreft hetinderdaad een belangrijke richting en wordt het reële nulpunt berekend. Deze nulpuntenworden gebruikt om het responsoppervlak aan te passen. Is de richting niet belangrijkgenoeg - de bijdrage tot de faalkans is klein - dan kan het resultaat van hetresponsoppervlak gebruikt worden. De winst ligt uiteraard in deze onrechtstreeksegrenstoestandsevaluatie.

De globale efficiëntie bedraagt ongeveer N=15n, waarmee het aantal evaluaties opnieuwproportioneel is met het aantal toevalsvariabelen. Er is geen voorkeur voor een welbepaaldefaalmode, zodat opnieuw een faalkans op niveau III wordt berekend. Deze techniek is verderuitgewerkt en de efficiëntie is verhoogd: • om het responsoppervlak te valideren wordt een F-test uitgevoerd,• een variabele additionele afstand λadd= var wordt ingevoerd in plaats van de constante

en arbitrair gekozen waarde λadd=3.0. De variabele additionele afstand is rechtstreeksfunctie van de fout tussen het responsoppervlak en het reële systeem,

• er wordt automatisch gekozen tussen een schare van lage orde veeltermen. Deze die leidttot de kleinste fout wordt aangewend als functioneel verband voor het responsoppervlak.

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Stap 3: Directional Sampling ophet responsoppervlak:

If λ i,RS < λmin+λaddBerekenpi(LSF)=χ2(λ i,LSF,n)Pas het respons-oppervlak aan met hetnieuwe nulpunt

ElseBerekenpi(RS)= χ2(λ i,RS,n)

u1

u2

gRS,2= 0.92+0.046u1 -0.023u2-0.074u1u2 -0.097u1

2-0.084u22

gRS,2 = 0

λmin = 2.05λadd = 3.0

Figuur 1: DARS (λadd = var) - Stap 3 in de betrouwbaarheidsprocedure

Stap 3 voor de aangepaste techniek met λadd=var, is weergegeven in Figuur 1 voor hetacademisch voorbeeld.

3.2. Monte Carlo en FORM met een aanpasbaar responsoppervlak (ARS)Hoofdstuk 2 gaf aan dat de efficiëntie van Monte Carlo met Variance Increase vergelijkbaar ismet Directional Sampling. De combinatie met een aanpasbaar responsoppervlak leidt dan ooktot een gelijkaardig resultaat.

De procedure verschilt enkel van de DARS procedure in de laatste stap. Deze stap wordt alsvolgt geherformuleerd:• Stap 3: deze stap is opnieuw een iteratieve stap. Variance Increase Monte Carlo

Sampling wordt toegepast op het responsoppervlak. De uitkomst op basis van ditresponsoppervlak wordt geëvalueerd. Valt het punt buiten het grensgebied tussen veiligen onveilig, dan kan dit resultaat gehanteerd worden. Omgekeerd, ligt het binnen hetinteressante gebied, dan wordt een nieuwe grenstoestandsfunctie geëvalueerd op hetoriginele systeem. De nieuwe data worden gebruikt om het responsoppervlak aan tepassen.

Voor de variantietoename wordt het experimentele verband uit Vgl. 6 gebruikt. Een eersteschatting wordt verkregen uit stap 1 in de procedure.

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u

gLSFRS

λλλλRS

λλλλadd

∆∆∆∆g,add

∆∆∆∆g,add

λλλλ i

gRS,i

gLSF,i

εεεεg,i

Figuur 2: Onderscheid tussen de belangrijke en niet-belangrijke regio in u-ruimte

Om na te gaan of het punt tot het grensgebied tussen veilig- en onveilig behoort, worden tweealternatieven uitgewerkt, Figuur 2. Een eerste is gestoeld op de DARS procedure, waarbij deafstand tot de grens tussen veilig en onveilig gebied wordt gehanteerd. Omdat bij de MonteCarlo techniek geen nulpunten worden berekend, is het gebruik van een variabele afstand nietvanzelfsprekend. Een arbitraire waarde (λadd) gelijk aan 3 wordt aangehouden. Een tweedemethode bestaat erin een afstand in de uitkomstruimte te definiëren die gerelateerd wordt aan defout in de uitkomstruimte:

(7)ε g i LSF i RS ig g, , ,= −

(8)( ) ( )( ){ }∆ g add g n gabs tLSFE, . ,max= ± ×−µ ε σ ε0 01 1

Hiermee wordt opnieuw een objectief criterium tot stand gebracht om het onderscheid te maken.

De efficiëntie van DARS en MCARS+VI hangen af van de mogelijkheid van hetresponsoppervlak om het reële systeemgedrag te beschrijven. Voor meer complexe systemen zalDARS efficiënter zijn. Dit is voornamelijk omdat DARS enkel nulpunten gebruikt om hetresponsoppervlak aan te passen. Dus enkel een beschrijving in het kritieke gebied tussen veiligen onveilig wordt gerealiseerd. MCARS, waar geen nulpunten worden berekend, maakt gebruikvan alle grenstoestandsevaluaties. Voor complexe systemen kan een lage orde veeltermonvoldoende zijn om het globaal gedrag voldoende nauwkeurig te beschrijven, waar hetbeschrijven van de nulpunten misschien nog wel voldoende nauwkeurig is.

Voor eenvoudige systemen daarentegen zal MCARS+VI efficiënter zijn. Dit juist omdat het alle

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LSFE gebruikt. Daardoor zal het responsoppervlak sneller aansluiten bij het reële gedrag.Weglaten van tussenliggende resultaten, zoals bij DARS, staat dan gelijk aan weglaten vaninformatie.

Omdat eerste orde betrouwbaarheidsmethodes (FORM) zeer efficiënt zijn, werden reeds eerderstappen ondernomen om ze te combineren met een aanpasbaar responsoppervlak. Groot nadeelblijft dat het een niveau II procedure is. Achteraf blijft een systeemanalyse vereist om debijdragen van de verschillende faalmodes te combineren. Dit levert slechts een onder- enbovengrens op voor de globale faalkans. Daarom wordt de techniek niet verder gebruikt.

3.3. Vergelijking met niveau I methodesUitgaande van de niveau III methode kunnen de controlepunten berekend worden die in eenniveau I methode (methode der grenstoestanden) worden gehanteerd, ter controle van deveiligheid:

(9)RS

d R T R

d S T S

= −= +µ β σµ β σ

0 80 7

..

Het actuele veiligheidsniveau wordt niet berekend. Er wordt enkel gecontroleerd (Rd<>Sd) of aaneen vooropgesteld niveau (βT) voldaan wordt.

Wanneer de grenstoestandsfunctie analytisch gekend is, zijn de meestebetrouwbaarheidsmethodes geschikt om een accurate waarde voor de faalkans te berekenen, zoook de traditionele methodes, beschreven in Hoofdstuk 2. Wanneer dit niet het geval is, en meernog, wanneer een evaluatie veel rekenwerk vereist, dan verdient een combinatie van DirectionalSampling of Monte Carlo met een aanpasbaar responsoppervlak de voorkeur.

4. Rekenmodellen voor metselwerk

Om de veiligheid te berekenen van een structuur, zijn steeds de grenstoestanden (uiterstegrenstoestand, gebruiksgrenstoestand) vereist. Dit ongeacht of de analyse uitgevoerd wordtovereenkomstig een niveau I, II dan wel niveau III methode. Dit hoofdstuk geeft een overzichtvan de verschillende modellen die voorhanden zijn en die ook gebruikt zullen worden in detoepassingen om het onderscheid tussen veilig en onveilig te bepalen.

Wat betreft de analytische modellen ligt de nadruk op metselwerk in druk. Omdat centraalaangrijpende drukkrachten haast nooit zullen voorkomen, wordt het effect van een excentrischebelasting in detail beschreven, Figuur 3. Naast een materiaalmodel dat geen trek kan opnemen(NTM), wordt de invloed van een beperkte treksterkte en plastisch gedrag in druk behandeld.

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d

FV

e

wVoor-aanzicht

Bovenaanzicht

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Excentriciteit e [mm]

e=d/6Dru

kspa

nnin

g σ

[N/m

m2 ]

FVe

fc

dc dpl

ElastischplastischmodelFVe

ft

dc

fcElastischbrosmodel

NTM

ft≠0

Figuur 3: Metselwerk belast op een excentrisch aangrijpende drukkracht

Dit deel sluit naadloos aan bij de beschrijving van bogen. Immers, bogen dragen verticale lastenaf door het vormen van een druklijn die de vorm van de fysische boog volgt. In een willekeurigedoorsnede leidt dit tot een excentrisch aangrijpende drukkracht. Het veiligheidstheorema vooreen boog luidt: “Als een druklijn kan gevonden worden die in evenwicht is met de uitwendigekrachten en die geheel binnen de fysische grenzen van de boog gelegen is, dan is de structuurveilig.”

Voor excentrische druk en boogwerking worden ter vergelijking de analytische modellen uit EC6opgenomen. Omdat deze een eerder conservatief resultaat opleveren, is een zekerevoorzichtigheid nodig wanneer ze gehanteerd worden in een probabilistische benadering. Hetis immers nodig dat de grens tussen veilig en onveilig zo juist mogelijk is. Het model zelf magniet intrinsiek al een zekere veiligheidsmarge bevatten. Dit beïnvloedt het resultaat.

Analytische modellen voor horizontale lasten zoals windlasten en afschuifwanden zijn eerderelementair. Zij tonen het belang aan van numerieke modellen om een voldoende nauwkeurigeovereenkomst met de realiteit te verkrijgen zoals nodig wanneer ze deel uitmaken van eengrenstoestandsfunctie bij een probabilistische benadering.

Bij de numerieke modellering gaat de aandacht voornamelijk naar eindige elementen modellen(FEM). Discrete elementen modellen (DEM) worden niet behandeld, al leveren ook zijveelbelovende resultaten op.

Hoewel een gedetailleerd micro-model meer nauwkeurige resultaten oplevert, wordt een macro-model voor metselwerk vooropgesteld. Dit om redenen van rekenefficiëntie. Bedoeling isimmers een methode te bewerkstelligen, haalbaar voor ganse structuren of structurele elementen.

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Rankinevon Mises

σ2

ft

ft

fc

fc

σ2RankineDrucker Prager

ft

ftfc

fc

σ1σ2

τ12

Rankine criterium

σ1

σ2

τ12

Hill criterium

σ2

σ1

ftft

fc

fc

τ0τ1 τ2 τ3

Diana model

Figuur 4: Rankine en Hill criterium in druk en trek

Een micro-model zou in dat geval onaanvaardbaar veel rekentijd vergen. Het globale anisotropecontinuum model opgesteld door Lourenço, wordt gehanteerd. Dit model omvat een combinatievan een Rankine vloeicriterium in trek en een Hill criterium in druk, Figuur 4.

Voor metselwerk in druk, trek en afschuiving zijn, naargelang het gebruikte eindige elementenpakket (Atena2D, Diana) verschillende mogelijkheden voorhanden, alsook beperkingengekoppeld. Een van de beperkingen is het veronderstelde isotrope materiaalgedrag.

De rekenmodellen vereisen de invoer van een hele reeks materiaaleigenschappen. Wanneer eenprobabilistische procedure wordt nagestreefd, dan zijn de waarschijnlijkheidsverdelingen vandeze materiaaleigenschappen vereist.

5. Experimenteel onderzoek

Dit hoofdstuk geeft de resultaten weer van het experimenteel onderzoek op metselwerk, decomponenten baksteen en mortel en de statistische verwerking van de data. Het doel is dewaarschijnlijkheidsverdelingen op te stellen voor de belangrijkste materiaalparameters die vereistzijn in de rekenmodellen die het onderscheid tussen het veilige en onveilige gebied maken. Hetonderzoek richt zich op drie niveau’s: de componenten baksteen en mortel, het composietmetselwerk en de relatie tussen beide. De belangrijkste elementen zijn de variabiliteit van hetcomposiet metselwerk (verdelingstype en parameters), de component-composiet relatie en debruikbaarheid van beschikbare testen om deze informatie in situ te verzamelen bij bestaandestructuren. Omdat het aantal proeven relatief groot is voor bepaalde testen, kunnen enkeleinteressante opmerkingen gemaakt worden omtrent grootte-effect, verschil tussen proefmethodesen anisotropie. Dit onderzoek betracht niet enkel experimenteel onderzoek. Daarom zullen, daarwaar nodig en mogelijk, de experimentele waarden aangevuld worden met literatuurgegevens.

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De variabiliteit van het metselwerk is slechts gedeeltelijk het gevolg van intrinsiekemateriaalvariabiliteit. Extra onzekerheid komt van de gebruikte proefprocedure, het beperkteaantal proefmonsters of de beperkte kennis van het fenomeen zelf. In vergelijking tot anderematerialen zoals staal of beton, is een extra onzekerheid te wijten aan het composietgedrag.Metselwerk is opgebouwd uit (bak)steen en mortel, volgens een (on)regelmatig patroon. Relatiefgrote proefmonsters zijn vereist om de materiaaleigenschappen van dit composiet te achterhalen.

Om de verschillende proefresultaten onderling te kunnen vergelijken, is slechts één typebaksteen, één type mortel en één type metselverband (Vlaams Verband) gebruikt. Het aantalproefmonsters is voldoende groot gehouden om de statistische onzekerheid te beperken en omonderling significante uitspraken te kunnen doen. Verschillende proeftechnieken wordengebruikt, Tabel 4.

Type proef geometrie Data [#proefstukken]

mortel standaardmortelbalk

3-puntsbuigproef drukproef

l×h×w = 160×40×40 mml×h×w = 40×40×40 mm

ft [53]fc, σ-ε, E [108]

kernen triaxiaaltest α = 0q,� = 150 mm, h = 300 mm vloeicriterium [28]

baksteen doubletten drukproef 2-lagen baksteen en een enkelevoeg, l×h×d = 188×126×48mm

fc, σ-ε, E [50]

kernen drukproef verticaal geboord uit baksteen�=50 mm, h =58 mm

fc, σ-ε, E [51]

prisma’s drukproef b×h×d = 160×45×25 mm fc, σ-ε, E [40]

metsel-werk

kernen drukproefafschuifproef

α = 0q, 45q� = 150 mm, h = 300 mm

fc, σ-ε, E, Gfc [5] τ-γ, GfII [5]

trekproef verticaal geboord uit doubletten� = 50 mm, h = 126 mm

ft [56]

triaxiaaltest α = 0q, 45q� = 150 mm, h = 300 mm

vloeicriterium [30,15]

pijlertjes drukproef 6-lagen baksteen (kruisverband)w×h×d = 188×338×188 mm

fc, σ-ε, E[20]

muurtjes drukproefafschuifproef

α = 0q, 45qw×h×d = 582×570×188 mm

fc, σ-ε, E, Gfc [3] τ-γ, GfII [3]

Tabel 4: Overzicht experimenteel proefprogramma

5.1. Componenten baksteen en mortelEen handgevormde gevelsteen module 50 type Kempenbrand (NBN B23-002, 1986)l×w×h=188×88×48mm3, wordt onderworpen aan 3 types verplaatsingsgestuurde (v=1mm/min)drukproeven: kernen ‰50 met een hoogte van 44 mm (h2/S=1), doubletten met een hoogte gelijkaan 120 mm en prisma’s, gezaagd uit een baksteen, met een hoogte gelijk aan 160 mm. Deresultaten zijn samengevat in Tabel 5.

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fc[N/mm2]

kernen ‰50, h=44mm doubletten, h=120mm prisma’s, h=160mm

xs [N/mm2]cov [%]nPDF

6.342.3737.451LN

5.161.563050LN

8.001.9324.240LN

Tabel 5: Drukproeven op baksteen

Een extreme-waarde-analyse uitgevoerd op de kernen ‰50, leert dat een Pareto-verdeling bestaansluit bij de proefresultaten. De zware staart is echter veroorzaakt door het opvullen van degroef in de baksteen met een standaardmortel. Dit beïnvloedt de resultaten. Deze hoge waardenzijn geen eigenschap van de baksteen. Daarom wordt geopteerd voor een lognormaal verdeling.Om na te gaan of het verschil tussen de gemiddelde druksterkte van de populatie (µ) significantis, wordt de techniek van de hypothesetesten gebruikt. De nulhypothese (H0) en de alternatieveeenzijdige hypothese (H1) luiden: H0:µ1=µ2, H1: µ1 >µ2. Een significantie niveau van 95% wordtaangehouden. Daaruit blijkt inderdaad een significant verschil tussen de drie types proeven. Deverklaring voor dit verschil moet gezocht worden in twee richtingen. Enerzijds speelt hetgrootte-effect mee. Eurocode 6 voorziet een normalisatiefactor (δ) om dit te compenseren: fb =δ×fb,m=0.82×6.34 = 5.20 N/mm2 voor kernen ‰50 en fb = 1.11×5.16=5.73 N/mm2 voordoubletten. Het verschil tussen deze genormaliseerde waarden is niet langer significant.Wanneer de resultaten vergeleken worden met de prisma’s die uit een baksteen werden gezaagden volgens de lengte-as (x-as) werden beproefd, dan speelt de oriëntatie blijkbaar een belangrijkerol. Ondanks de grotere slankheid, zijn deze proefstukken sterker volgens deze materiaalas. Dekernen ‰50 en doubletten worden beproefd volgens de hoogteas van de baksteen (y-as), net zoalsde metselwerk proefstukken. Voor de druksterkte worden deze waarden verder gehanteerd.

De mortel is een kalk-cement mortel type M4 (NBN B24-301, 1980). De proefresultaten opstandaard mortelbalkjes zijn samengevat in Tabel 6. De mortelbalkjes werden aangemaaktsamen met het metselen van de andere proefstukken om zoveel mogelijk representatief te zijn.

195 dagen ft [N/mm2] fc,y [N/mm2] E [N/mm2]

xs [N/mm2]cov [%]nPDF

2.600.491953trunc. N/LN

8.311.8021.6108trunc. N/LN

139825218107trunc. N

Tabel 6: Mortel proefstukken l×w×h=40×40×160 mm3 - samenvattende statistiek van deproefresultaten

5.2. Composiet metselwerkDe eerste serie proefstukken zijn kleine pijlertjes, opgebouwd uit 6 lagen, Figuur 5. Deresulterende materiaaleigenschappen - fc, Ey, Gfc - zijn weergegeven in Figuur 5. De gemiddeldedruksterkte is lager dan de waarden gevonden op de baksteen- en mortelproefstukken. De

-xxxiv-

0

1

2

3

4

5

6

0 0.003 0.005 0.008 0.01 0.013 0.015

Rek ε [mm/mm]

σ

140dagen

fc,y[N/mm2]

Ey[N/mm2]

Gf,c[Nmm/mm2]

xscov [%]nPDF

4.260.831919(N)/LN

16733682218N

1.710.563318N

h =

360

mm

w = 188 mm d = 18

8 mm20

0 m

m

LVDT

Figuur 5: Metselwerk pijlertjes - spanning-rek relatie en materiaaleigenschappen

0

1

2

3

4

5

6

0 0.003 0.005 0.008 0.01 0.013 0.015

Rek ε [mm/mm]

σ

152dagen

fc,y[N/mm2]

Ey[N/mm2]

Gf,c[Nmm/mm2]

xscov [%]n

4.540.77175

1690372225

1.351.02765

200

mm

h =

300

mm

φ = 150 mm

LVDT

Figuur 6: Metselwerk kernen - spanning-rek verband en materiaaleigenschappen

waarden van de breukenergie stemmen overeen met literatuurwaarden.

De tweede reeks proefstukken bestaat uit 5 metselwerk kernen, diameter 150mm en hoogte300mm. Deze werden uit muurtjes geboord die gelijktijdig met de andere proefstukken zijnopgetrokken. De materiaaleigenschappen - fc, Ey, Gfc - zijn mee opgenomen in Figuur 6. Globaalgenomen zijn de gemiddelde waarden sterk gelijklopend met de waarden op pijlertjes, Figuur 5.Het aantal proeven is te klein om een verdelingstype uit af te leiden.

Drie metselwerk muurtjes zijn het voorwerp van de laatste reeks proefstukken, Figuur 7:w×d×h=600×188×600 mm3. De gemiddelde druksterkte is relatief hoog (fc = 6.07 N/mm2). De

-xxxv-

0

1

2

3

4

5

6

7

8

0 0.003 0.006 0.009 0.012 0.015Rek ε [mm/mm]

σ

141dagen

fc

[N/mm2]Ey[N/mm2]

Y Gfc[Nmm/mm2]

x 6.07 1642 0.19 2.73

s 0.57 140 0.06 0.60

Cov [%] 9.3 8.5 31 22

h =

570

mm

w = 577 mmd =

188 m

m

lref = 400 mm

l ref =

460

mm

lref = 100 mm

Figuur 7: Metselwerk muurtjes - spanning-rek relatie en materiaaleigenschappen

spreiding op de druksterkte is klein. Dit kan aangeven dat grotere proefstukken tot eenhomogener geheel leiden. De elasticiteitsmodulus (E=1642 N/mm2) valt vrijwel samen met dewaarde opgemeten op pijlertjes (E=1673 N/mm2) en metselwerk kernen (E=1690 N/mm2).

Op basis van de horizontaal geplaatste LVDT’s wordt een waarde voor de coëfficiënt vanPoisson (ν) berekend. De spreiding is relatief hoog, wat mee aangeeft dat deze parameter nietzo eenvoudig kan worden opgemeten als de andere materiaaleigenschappen. Toch stemt degemiddelde waarde goed overeen met wat in de literatuur wordt teruggevonden. De gemiddeldebreukenergie (Gfc=2.73 Nmm/mm2) is hoger dan in het geval van metselwerk pijlertjes (Gfc=1.71N/mm2). De metselwerk muurtjes hebben niet enkel een minder bros breukgedrag, de hogerebreukspanning draagt evenzeer bij tot de hogere waarden voor de breukenergie.

De gemiddelde druksterkte blijkt af te nemen in functie van de slankheid van het proefstuk,Figuur 8. De waarden verkregen uit pijlertjes en kernen blijken daarbij meer representatief te zijnvoor metselwerk dan metselwerk muurtjes. Omdat kernen eenvoudiger te ontnemen zijn, wordtdit type proefstukken verkozen. Bijkomend geven ze informatie over de reële toestand van hetgebouw.

-xxxvi-

0

1

2

3

4

5

6

7

8

1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65Slankheid: λ=

muurtjes (n=3)

kernen φ=113 mm, h=170 mm (n=6)(*)

pijlertjes (n=19)

kernen φ=150mm (n=5)

muur (n=1)(*): wxhxd=2000x2000x188

( )λ ρ= 2h

d wef

EC6, 1995

Figuur 8: Druksterkte in functie van slankheid (*: literatuurwaarden)

5.3. Metselwerk in druk - stochastische uitbreidingOnder de vele andere formules, is het verband dat wordt voorgesteld in EC6 (1995) een van demeest betrouwbare en praktisch bruikbare om de druksterkte van het composiet metselwerk (f’k)af te leiden uit de sterktewaarden van de componenten baksteen (fb) en mortel (fm):

(10)( ) ( )f K f fk b m' . ..= δ 0 65 0 25

Voor de baksteen wordt de genormaliseerde druksterkte gehanteerd (δ). De waarde voor decoëfficiënt K voor groep 1 metselwerk stenen bedraagt 0.6. De gemiddelde waarde en spreidingvan de resulterende lognormale verdeling kan analytisch berekend worden, Tabel 7.

fc [N/mm2] PDF µ σ cov proefstukken

metselwerkexperimenteel

LN 4.26 0.81 19 pijlertjes

/ 4.53 0.77 17 kernen ‰150

metselwerknumeriek,Vgl. 10

LN 4.60 1.00 22 baksteen sterkte op basis van kernen ‰50

LN 4.55 0.84 18 baksteen sterkte op basis van doubletten

Tabel 7: Waarschijnlijkheidsverdeling en parameters voor metselwerk druksterkte

De numerieke resultaten (fc=4.55-4.60 N/mm2) stemmen overeen met de experimentele (fc=4.26-4.53 N/mm2). Tevens wordt een goede waarde voor de variatiecoëfficiënt verkregen. In beidegevallen schommelt deze rond de 20%. Het numerieke model bevestigt het idee dat een meer

-xxxvii-

Homogenisering volgens x-as

Homogeniseringvolgens y-as

x

y

B1 B2 B3

B4 B5 B6

B7 B8 B9

M1 M2

M3M4 M5

M6M7 M8

d4

Figuur 9: Homogenisering - basiscel en twee-stappen-x-y-homogeniseringsprocedure voorperiodieke gelaagde materialen

homogeen composiet wordt gevormd, wanneer de variatiecoëfficiënt vergeleken wordt met dezevan de componenten baksteen en mortel.

5.4. Homogenisering - stochastische uitbreidingDe homogeniseringsprocedure zoals voorgesteld door Lourenço, wordt uitgebreid met deinvoering van toevalsvariabelen voor materiaaleigenschappen en geometrie. Dit om na te gaanof deze numerieke procedure een volwaardig alternatief oplevert voor het rechtstreeks schattenvan de waarschijnlijkheidsverdeling van de elasticiteitsmodulus (E) op metselwerk proefstukken.Bedoeling is om de waarschijnlijkheidsverdeling van de elasticiteitsmodulus af te leiden uit decomponentwaarden.

De procedure bestaat hoofdzakelijk uit twee opeenvolgende homogeniseringsstappen voorperiodieke gelaagde materialen. Elke component wordt verondersteld isotroop te zijn. Degehomogeniseerde stijfheidsmatrix (Ch) wordt verkregen uit de constitutieve eigenschappen vande componenten (Ci) baksteen en mortel, overeenkomstig:

(11)( ) ( )C P C P P C P Chi t i e

ii t i e

iip p= −� � −

− −1 1

Hierin zijn Pt en Pe projectie matrices en pi de relatieve dikte van een laag in het periodiekmedium. De basiscel voor metselwerk heeft duidelijk geen gelaagde structuur. Verschillendeauteurs suggereren een benaderende aanpak met een twee-stappen-procedure, Figuur 9.

Om de fysische onzekerheden van baksteen en mortel in rekening te brengen, wordt voor beidede elasticiteitsmodulus als toevalsvariabelen in rekening gebracht. Om ook de geometrischeonzekerheid in rekening te brengen wordt de dikte van de mortel laag als toevalsvariabelegedefinieerd. De aangehouden parameters voor deze toevalsvariabelen zijn samengevat in Tabel8. Zij zijn afkomstig van het experimenteel onderzoek. Om de resulterende stijfheidsmatrix teberekenen wordt een Monte Carlo simulatie (N=105) gebruikt bij de twee-stappen-x-y-homogeniseringsprocedure. De resultaten zijn samengevat in Tabel 9.

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0

0.5

1

1.5

2

2.5

3

3.5

0 0.005 0.01 0.015 0.02 0.025

Afschuifhoek γ [rad]

τ

200

mm

h =

300

mm

d = 150 mm

LVDT

209dagen

gmax[N/mm2]

G[N/mm2]

Gf,II[Nmm/mm2]

xscov [%]n

2.430.43185

883271315

0.860.57665

Figuur 10: Metselwerk kernen - τ-γ verband en materiaaleigenschappen

Toevalsvariabelen PDF µ cov [%]Baksteen (prisma’s) Eb1,...,Eb9 LN 1711 N/mm2 23Mortel (mortel balkjes) Em1,...,Em8 N 1398 N/mm2 18

dm1,...,dm8 LN 10 mm 10Tabel 8: Toevalsvariabelen voor de componenten baksteen, mortel en de geometrie

elasticiteitsmodulus (Ey) µ [N/mm2] σ [N/mm2] cov [%] # PDF

kernen φ150 1690 372 22 5 /

muurtjes V1-V3 1642 140 8,5 3 /

volle schaal muur 1587 / / 1 /

pijlertjes 1673 368 22 18 /

xy-homogenisering 1596 119 7,5 105 NTabel 9: Elasticiteitsmodulus, vergelijking tussen experimentele en numerieke waarden

Het is duidelijk dat de resulterende stijfheid ergens tussen deze van de baksteen (E=1717 N/mm2)en mortel (E=1398 N/mm2) in moet gelegen zijn. Hoewel de componenten zelf een redelijk grotespreiding vertonen, is de spreiding op het resulterende metselwerk klein (cov(Ey)=7.5%).Opnieuw wordt een meer homogeen resultaat verkregen.

5.5. Metselwerk in afschuivingDe eerste reeks proefstukken zijn kernen, geboord uit muurtjes onder een hoek van 45°. Deschuifsterkte (τmax), glijdingsmodulus (G) en breukenergie (Gf,II) voor een mode II afschuifbreuk,zijn weergegeven in figuur 10.

-xxxix-

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.000 0.005 0.010 0.015 0.020 0.025Afschuifhoek γ [rad]

τ

158 dagen gmax [N/mm2] G [N/mm2] Gf,II [Nmm/mm2]x 1.49 748 1.32

s 0.19 108 0.76

cov [%] 13 14 58

l0 = 300 mm l0 = 300 mm

l 0 =

300

mm

w= 577 m

m

h= 570 mm

w = 300 mm

Figuur 11: Metselwerk muurtjes - τ-γ verband en samenvattende resultaten

Het tweede type proefstukken zijn drie metselwerk muurtjes, Figuur 11. De glijdingsmodulusleunt sterk aan bij de waarde die numeriek kan berekend worden: G=E/2(1+ν). Wanneer Ey=1642 N/mm2 en ν=0.19, Figuur 7, dan is de glijdingsmodulus gelijk aan: G= 690 N/mm2. Dit isvergelijkbaar met het experimenteel gemiddelde: (G)= 748 N/mm2. x

Literatuurwaarden voor de breukenergie (Gf,II) liggen veelal lager (0.05-0.50 Nmm/mm2). Zekerwanneer de hechting faalt, ligt de breukenergie laag. Bij dit type metselwerk is een grotehaakweerstand aanwezig omdat de bakstenen een groef bevatten. De mortelvulling ervanveroorzaakt een hoge wrijvingsweerstand in het afschuifoppervlak. Omwille van dehaakweerstand, wordt een hoge spreiding verkregen in het nascheurgedrag, wat merkbaar is inGf,II.

De wrijvingswet van Coulomb wordt vaak gebruikt in analytische modellen: τ=c+σ×tan(φ). Dewrijvingscoëfficiënt is daarbij relatief onafhankelijk van het type baksteen of mortel dat gebruiktwordt: tan(φ)~LN(0.81,(0.15)2). Daarmee kan de cohesiecoëfficiënt (c) berekend worden,gebruik makend van de proefresultaten (τmax): c~LN(0.50;(0.15)2).

5.6. Metselwerk in trekVoor het bepalen van de treksterkte van metselwerk, wordt een directe trekproef gebruikt (prEN-1542): ft~LN(0.28,(0.10)2). Van de 56 proefstukken, begaven 8 in de baksteen/mortelovergangszone, 10 in de mortel en 38 in de baksteen zelf. De breukenergie (Gf,I) wordt berekendop basis van de dalende tak in de spanning-scheuropening-relatie (σ-wc):

(12)σf

et

fG

wf

f Ic

=−

,

Een schatting van de breukenergie wordt verkregen op basis van lineaire regressie:Gf,I~LN(0.005,(0.0033)2).

-xl-

triaxiaalcelRing met radiale LVDT’s op 90°Axiale LVDT’s op 90°Mortel proefstuk in mantel

Data-acquisitieTitan hoge druk pompINFCP

Figuur 12: Triaxiaalcel - proefopstelling

5.7. Metselwerk in een multi-axiale spanningstoestandSpeciaal voor de studie van het materiaalgedrag van heterogene materialen met lage sterkte, werdeen triaxiaalcel aangekocht door het Laboratorium Reyntjens, Figuur 12. De studie richt zich opde wijzigingen in het materiaalgedrag wanneer het onderhevig is aan een multi-axialespanningstoestand. Deze wordt in de triaxiaalcel gesimuleerd door een steundruk en eenverticale spanning te laten aangrijpen op geboorde metselwerk kernen ‰150 mm, met hoogte tot300 mm. Specifiek gaat de aandacht naar het plastisch gedrag, grotere vervormbaarheid en deoptredende versterkingsfactor bij de breuksterkte.

De breukresultaten voor de mortelproefstukken (‰150 en h=120mm) zijn weergegeven in Figuur13. De samenstelling van de mortelkernen is identiek aan de mortelbalkjes, zie hoger. Naast debreukwaarden zijn tevens de spanningspaden aangegeven en drie mogelijke modellen om hetvloeicriterium en de versterkingsfactor ten gevolge van de steundruk te schatten.

-xli-

02468

1012141618202224

0 2 4 6 8 10 12 14 16 18 20 22 24

Legende :♦ gefaald niet gefaald

Belastingspadmet constanteσ3/σ1-verhouding

σ1(=σV) [MPa]

σ3=σ2 (=σc) [MPa]

Lourenço’s model

Elliptisch vloeicriterium

Vloeicriterium inpoolcoördinaten, kleinstekwadraten benadering

0.050.10

0.15

0.25 0.50 0.75 1.00

Figuur 13: Triaxiaalproeven op mortel - resultaten en vloeicriterium

Reeds bij lage verhoudingen (σ3/σ1=0.05) wordt een overgang van quasi-bros naar elastisch-plastisch materiaalgedrag vastgesteld, alsook een sterke toename in (plastische) vervorming.Wanneer wordt uitgegaan van een maximale steundruk in de mortelvoeg gelijk aan de treksterktevan de baksteen (ft=0.28 N/mm²), dan wordt een spanningsverhouding σ3/σ1=0.043 bereikt.Onder deze voorwaarde is plastische vervorming van de mortelvoeg plausibel. Tot slot wordenreeds bij deze lage steundrukken beduidend hogere druksterktes opgemeten.

Voor de metselwerkproefstukken (‰150 en h=300mm) zijn de resultaten minder talrijk, zeker bijde hogere σ3/σ1-verhoudingen. Het optreden van lekken in de oliedichting omheen deproefstukken en het bereiken van de maximaal toelaatbare steundruk zijn de voornaamsteoorzaken. Hoewel het niet mogelijk was om op basis van de verkregen proefresultaten hetanisotroop gedrag van het metselwerk te begroten, noch het volledige vloeicriterium in de multi-axiale druk-druk-zone, kon de versterkingsfactor bij lage steundrukken bepaald worden. Tevenswerd dezelfde overgang naar plastisch materiaalgedrag teruggevonden bij verhogingen van deσ3/σ1-verhouding: σ3/σ1A0.4.

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De belangrijkste materiaaleigenschappen, hun verdelingstype en parameters zijn samengevat inTabel 10.

materiaaleigenschappen Eenheid µ σ PDF

elastischeeigenschappen

E [N/mm2] 1600 120 N

ν [Nmm/mm2] 0,19 0,06 LN

G [N/mm2] 748 108 LN

druk fc [N/mm2] 4,5 0,85 LN

Gfc [Nmm/mm2] 2,73 0,6 LN

afschuiving c [N/mm2] 0,5 0,15 LN

tan(φ) / 0,81 0,15 LN

Gf,II [Nmm/mm2] 1,32 0,76 LN

trek ft [N/mm2] 0,28 0,1 LN

Gf,I [Nmm/mm2] 0,05 0,03 LNTabel 10:Metselwerk materiaaleigenschappen, verdelingstype en parameters

6. Toepassingen

Dit hoofdstuk bevat vier voorbeelden waarin de methodologie wordt toegepast op ongewapendmetselwerk. Niet alleen analytische problemen komen aan bod, zoals grout-injectie vanmeerschalig metselwerk of een metselwerk kolom belast op een excentrische drukkracht. Meercomplexe metselwerkstructuren zoals metselwerk bogen en afschuifwanden illustreren decombinatie van de probabilistische procedure met een externe numerieke methode die degrenstoestand evalueert. In de laatste drie voorbeelden wordt het niet-lineair materiaalgedrag inrekening gebracht. Met de eerste voorbeelden ligt de nadruk op historisch metselwerk. Met hetlaatste voorbeeld verschuift de nadruk naar meer hedendaagse toepassingen. Daar waar mogelijkwordt de vergelijking gemaakt met het veiligheidsniveau volgens een niveau I procedure ofmethode der grenstoestanden.

6.1. Consilidatie van meerschalig metselwerkMeerschalig metselwerk komt veelvuldig voor bij historische gebouwen. Wanneer hetdraagvermogen van deze wanden in vraag wordt gesteld, kan een groutinjectie oplossing bieden.De druksterkte na injectie kan berekend worden op basis van de bijdrage van de externeparementen (Vext, fext) en die van het geïnjecteerde breuksteenmetselwerk ertussen (Vinf, finf,s):

(13)f VV

fVV

fwc sext

ext s,inf

inf,= � � + ���

��

�

��

��ε

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Groutinjectie: initieel holtepercentage V0 = 30%

33.23.43.63.8

44.24.44.64.8

5

2 4 6 8 10 12 14 16Gemiddelde druksterkte injectiemateriaal: µ(fgr)

β

V(fgr)=0,10V(fgr)=0,15V(fgr)=0,20V(fgr)=0,25

variatiecoëfficiënt vanhet injectiemateriaal

βT=3,7

Vinj=20%Vinj=25%

Vinj=30%

Figuur 14: Evolutie van de betrouwbaarheidsindex bij consolidatie van meerschalig metselwerk

De modelonzekerheid (ε) wordt ingevoerd omdat het verband opgesteld is op basis van eenbeperkt aantal experimenten. De sterkte van het geïnjecteerde breuksteenmetselwerk op haarbeurt kan verkregen worden uitgaande van de sterkte van de grout (fgr) via:

(14)f finj grinf,..= 0 31 1 18

De parameters van de toevalsvariabelen zijn samengevat in Tabel 11.

Toevalsvariabelen parameter µ σ cov [%] PDF

verticale spanningbuitenparementopvulmetselwerkgroutvolume holtengeïnjecteerd volumemodelonzekerheid

S [N/mm2]fext [N/mm2]finf [N/mm2]fgr [N/mm2]V0 [%]Vinj [%]ε

1.04.260.5112.0530var1

0.070.810.401.04//0.15

719789//15

NLNLNLNconstantconstantN

Tabel 11: Toevalsvariabelen en hun parameters

Op basis van deze analyse is het mogelijk om evaluatiecurven op te stellen ter controle van deuitgevoerde injectie. Figuur 14 geeft de betrouwbaarheidsindex weer in functie van devullingsgraad en de materiaalparameters van het injectieproduct.

Hieruit blijkt dat niet enkel de gemiddelde druksterkte van de grout van belang is, maar tevens

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de spreiding op deze materiaalkarakteristiek. Dit benadrukt het belang van een verzorgdeuitvoering.

6.2. Een kolom belast op een excentrisch aangrijpende drukkrachtBij de beoordeling van de veiligheid (uiterste grenstoestand) van een metselwerk kolom, belastop een excentrische drukkracht, wordt zowel een elastisch-bros (overeenkomstig NBN B24-301,1980) alsook een elastisch-plastisch materiaalmodel (overeenkomstig EC6, 1995) geanalyseerd,Figuur 3. De winst in veiligheid wordt bekeken. Daarnaast wordt ook de invloed van eenbeperkte treksterkte in rekening gebracht bij het optreden van scheurvorming(gebruiksgrenstoestand).

Voor de niveau III methode worden zowel de traditionele methodes, Hoofdstuk 2, als degecombineerde technieken uit Hoofdstuk 3 gebruikt. Dit laat toe hun efficiëntie onderling tevergelijken. De toevalsvariabelen zijn samengevat in Tabel 12.

Toevalsvariabelen parameter µ σ cov [%] PDF

verticale krachtexcentriciteitbreedtediktedruksterktetreksterkte

FV [N]e [mm]w [mm]d [mm]fc [N/mm2]ft [N/mm2]

1400001506006004.260.28

140002530300.810.11

1017551939

NNNNLNLN

Tabel 12: toevalsvariabelen en hun parameters

De resultaten van de betrouwbaarheidsanalyse zijn weergegeven in Tabel 13. De impact van eenmeer gedetailleerd materiaalmodel laat zich gevoelen in de betrouwbaarheidsindex. Niet enkelvoor de uiterste grenstoestand (elastisch-bros Î elastisch-plastisch): β=3.66 Î 4.06 of pf = 1.310-4 Î 2.5 10-5, maar ook voor de gebruiksgrenstoesand (NTM Î ft £0): β=-1.96 Î 0.52 of pf= 0.98 Î 0.30. De veiligheid neemt in beide gevallen een orde grootte toe. Door het beterbegrijpen van het structureel gedrag, zal de betrouwbaarheidsindex verder convergeren naar dereële waarde. Oudere structuren, ontworpen op basis van een elastisch-bros of NTMmateriaalgedrag, neigen dan ook de veiligheid eerder conservatief te benaderen.

Ondanks het sterke niet-lineaire verloop van de uiterste grenstoestand in functie van deexcentriciteit (e), waarvoor een 2de orde veelterm niet het optimaal functioneel verband is, zijnde DARS en MCARS+VI procedures in staat om de faalkans accuraat te schatten. Omdat degrenstoestandsfuncties analytisch beschikbaar zijn, leveren de traditionele methodes zoalsFORM/SORM een vergelijkbare efficiëntie. Daar waar het centrum van het responsoppervlak(gemiddelde waarden voor de toevalsvariabelen) zich reeds in het faalgebied bevindt, biedtDARS/MCARS geen uitkomst. Dit doet zich voor bij het NTM materiaalmodel in degebruiksgrenstoestand.

Uiterste grenstoestand - elastisch-bros materiaalmodel

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niveau III procedure β pf #LSFE

nauwkeurigheid

DARS (λadd=var)MCARS+VI (λadd=3.0)MCARS+VI (λadd=var)

3.663.663.66

1.24 10-4

1.24 10-4

1.36 10-6

2482332

V(pf)=51%, σ(β)=0.17, V(β)=0.04V(pf)=51%, σ(β)=0.17, V(β)=0.04V(pf)=42%, σ(β)=0.17, V(β)=0.04

Uiterste grenstoestand - elastisch-plastisch materiaalmodel

DARS (λadd=var)MCARS+VI (λadd=3.0)MCARS+VI (λadd=var)

4.164.064.08

1.60 10-5

2.44 10-5

2.25 10-5

1112923

V(pf)=67%, σ(β)=0.09, V(β)=0.02V(pf)=67%, σ(β)=0.09, V(β)=0.02V(pf)=67%, σ(β)=0.09, V(β)=0.02

Gebruiksgrenstoestand - NTM materiaalmodel

FORM/SORMDARS/MCARS

-1.96/

0.98/

23/

//

Gebruiksgrenstoestand - ft £0

DARS (λadd=var)MCARS+VI (λadd=3.0)MCARS+VI (λadd=var)

0.520.480.55

0.300.320.29

16029028

V(pf)=9%, σ(β)=0.42V(pf)=9%, σ(β)=0.42V(pf)=9%, σ(β)=0.42

Tabel 13: Resultaat betrouwbaarheidsanalyse voor de verschillende grenstoestanden

Wanneer een probabilistische benadering wordt gebruikt, wordt de impact van elketoevalsvariabele begroot. Deze informatie is niet beschikbaar van een analyse volgens demethode der grenstoestanden (niveau I). Daarin worden veralgemeende invloedsfactorenaangenomen voor de verschillende probleemvariabelen.

6.3. Metselwerk bogenDit voorbeeld illustreert de praktische toepasbaarheid van de DARS en MCARS+VI methodevoor het geval de grenstoestandsfunctie enkel impliciet beschikbaar is. De automatischekoppeling tussen de betrouwbaarheidsprocedure en de grensanalyse via Calipous maken hetmogelijk de ganse procedure in een gesloten lus uit te voeren. Er zijn geen verdere manuelehandelingen vereist. De grensanalyse Calipous berekent de veiligheidsfactoren (αg en αS) voorde faalcriteria van de metselwerk boog op basis van een NTM-model en enkele geometrischeparameters, Tabel 14, Figuur 15.

Toevalsvariabelen Parameter µ σ cov [%] PDF

binnenstraal boogdikte boogafwijking t.o.v cirkelboog verticale kracht

x1 = r0 [m]x2 = d [m]x3= dr [m]x4 = FV [N]

2.500.160.0750

0.020.020.02150

0.040.17/20

NNNLN

Tabel 14: Toevalsvariabelen en parameters

Deze kunnen eenvoudig vergaard worden via een landmeetkundige opmeting. Op basis daarvan

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FV~LN(750,(150)²)

d~N(0.16,(0.02)²)

50 blokkenr=r0+drr0~N(2.5,(0.02)²)dr~N(0,(0.02)²)

Figuur 15: Metselwerk boog - definitie toevalsvariabelen

kan een eerste schatting gemaakt worden van het aanwezige veiligheidsniveau.

Omwille van de twee impliciete faalmodes, zijn DARS en MCARS+VI bijzonder geschikt omde systeemfaalkans te begroten, Tabel 15. Bij het gebruik van FORM/SORM zou de oplossingslechts convergeren naar een componentbetrouwbaarheid. Een systeemanalyse is dan vereist omuit deze componentbetrouwbaarheden een benaderende systeembetrouwbaarheid af te leiden.

Wanneer de huidige veiligheid onvoldoende blijkt, laat een sensitiviteitsanalyse in het meestwaarschijnlijke faalpunt (u*) toe om het effect van mogelijke interventies te beoordelen. In ditgeval is de dikte (d) de variabele met de grootste invloed op de veiligheid. Een toename van deveiligheid moet dan ook gezocht worden in een toename van de gemiddelde dikte of een afnamevan de spreiding op de dikte, Tabel 15. Dit eerste betekent een ingreep in de structuur, dit laatstebetekent een nauwkeurigere opmeting. Het responsoppervlak kan gebruikt worden om snel eeneerste schatting van de winst in veiligheid te begroten. Een toename in de nauwkeurigheid vande opmeting (σ(d) = 0.005 i.p.v 0.02 m) leidt tot een toename in betrouwbaarheid van 1.26 naar3.4-3.55, wat nagenoeg gelijk is aan de algemene streefwaarden, Hoofdstuk 1. Wanneer ditvoldoende wordt geacht, kan de structuur onaangeroerd blijven. Indien niet, dan is eenverhoogde gemiddelde dikte tot 0.21 m nodig om de betrouwbaarheid te verhogen tot 3.67-3.83.

Het gebruik van een variabele additionele afstand λadd=var, bewijst een zeer bruikbareverbetering te zijn in de praktijk. Het aantal grenstoestandsevaluaties daalt beduidend tegenoverde DARS methode met een constante afstand λadd=3.0. In dit voorbeeld is de winst in rekentijdvoelbaar. Een grensanalyse met Calipous vergt ongeveer 20 seconden. De winst is groternaarmate het responsoppervlak beter in staat is om het reële gedrag van de structuur te benaderen.Het feit dat doorheen de verschillende analyses, de waarde λadd varieert tussen 0.05-1.60,bevestigt dit. Daarnaast kiest de betrouwbaarheidsanalyse zelf welk functioneel verband uit eenfamilie van lage orde veeltermen, wordt verkozen. In sommige analyses is dit een zuiver

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kwadratisch model, in andere zijn ook de lineaire- en kruistermen aanwezig.

procedure d~N(0.16,(0.02)²)

β pf �LSFE(CPU-time)

nauwkeurigheid

DARS (λadd= 3.0)DARS (λadd=var)MCARS+VI (λadd=3.00)MCARS+VI (∆g,add=var)

1.261.221.251.30

0.110.120.110.10

149 (50 min)43 (17 min)45 (18 min)23 (12 min)

V(pf)= 33%, σ(β)=0.15V(pf)= 33%, σ(β)=0.15V(pf)= 33%, σ(β)=0.15V(pf)= 33%, σ(β)=0.15

betrouwbaarheidsanalyse - verhoogde nauwkeurigheid op de dikte: d~N(0.16,(0.005)²)

DARS (λadd= 3.0)DARS (λadd=var)MCARS+VI (λadd=3.00)MCARS+VI (∆g,add=var)

3.443.553.433.46

2.1 10-4

1.9 10-4

2.1 10-4

2.7 10-4

122 (41 min)34 (13 min)60 (22 min)56 (21 min)

V(β)=0.05V(β)=0.05V(β)=0.05V(β)=0.05

betrouwbaarheidsanalyse - verhoogde gemiddelde dikte: d~N(0.21,(0.02)²)

DARS (λadd= 3.0)DARS (λadd=var)MCARS+VI (λadd=3.00)MCARS+VI (∆g,add=var)

3.723.693.833.67

1.0 10-4

1.1 10-4

0.6 10-4

1.2 10-4

126 (52 min)45 (17 min)34 (14 min)23 (12 min)

V(β)=0.05V(β)=0.05V(β)=0.05V(β)=0.05

Tabel 15: Betrouwbaarheidsanalyse - initiële toestand en mogelijkheden voor verhoogdebetrouwbaarheid

Een gelijkaardige gedachtegang gaat op voor MCARS+VI. Het toevoegen van een variabeleafstand in de uitkomstruimte (∆g,add=var) leidt ook hier tot een verbetering. Hetresponsoppervlak valt nagenoeg samen met het reële structurele gedrag. De uiteindelijkevariabele afstanden zijn zeer beperkt en variëren tussen: ∆g,add= 0.016-0.09.

6.4. Metselwerk afschuifwandenAfschuifwanden hebben als functie het opnemen van horizontale lasten. Deze komen meestalvan de windbelasting op het gebouw. In dat geval zijn ze beperkt in vergelijking tot de verticalelasten. In aardbevingsgevoelige gebieden kunnen ze geïnduceerd worden door grondbewegingen.Dan kunnen ze de dominante belasting worden.

In de uiterste grenstoestand, gaat de interesse bij afschuifwanden voornamelijk naar het multi-modaal faalgedrag (verbrijzelen van metselwerk, trekbreuk en afschuifbreuk). Om dit gedrag opeen voldoende realistische manier te beschrijven volstaat het niet langer een linear-elastischmateriaalgedrag te hanteren. Het materiaalmodel moet de spanningsherverdeling na het optredenvan de eerste trekscheuren kunnen simuleren. Om rekening te houden met het fysisch niet-lineairgedrag, wordt de analyse uitgevoerd met het eindige elementen pakket Atena2D.

De toevalsvariabelen zijn samengevat in Tabel 16. De initiële verticale verplaatsing is eenvertaling van de invloed van het eigengewicht van de bovenliggende structuur.

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0

20

40

60

80

100

120

140

0 1 2 3 4 5 6 7Horizontale verplaatsing dx [mm]

Horizontale verplaatsing voor gemiddelde horizontale kracht

FH,max = 117,7 kN

X

Y

X

Y

dxFH

Optreden eerste scheur

71,7

45

0,56

Figuur 16: Kracht-verplaatsingsverloop voor gemiddelde waarden materiaalparameters

Toevalsvariabelen Parameter µ σ cov [%] PDF

S

R

x1x2x3x4x5x6x7

Horizontale krachtVerticale verplaatsingE-moduluscoëfficiënt van Poissondikte van de wanddruksterktetreksterkte

FH [kN]dy [m]E [N/mm2]ν [-]d [m]fc [N/mm2]ft [N/mm2]

456.04 10-4

16420.190.1884.260.28

11.255.4 10-5

1400.060.0080.810.10

2598320.41936

LNLNNLNNLNLN

Tabel 16: Toevalsvariabelen en hun parameters

Het kracht-verplaatsingsdiagramma voor de gemiddelde materiaalparameters en de optredendeschade zijn weergegeven in Figuur 16.

Uit de resultaten van de betrouwbaarheidsanalyses, Tabel 17, blijkt dat niet voldaan is aan destreefwaarden voor de uiterste grenstoestand: β=3.55<3.70. Elke grenstoestandsevaluatie vereisteen oproep naar het programma Atena2D. Aangezien een automatische koppeling nietrealiseerbaar is, vergt dit telkens een beperkte manuele handeling van de gebruiker. Omdat deduurtijd van een analyse ongeveer 10 minuten bedraagt, is het minimaliseren van het aantaldirecte evaluaties noodzakelijk. De keuze voor de DARS procedure met λadd=var ligt dan ookvoor de hand. De evolutie van de betrouwbaarheidsindex (β) alsook de additionele afstand λadd,zijn weergegeven in Figuur 17.

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2,00

2,20

2,40

2,60

2,80

3,00

3,20

3,40

3,60

3,80

4,00

0 100 200 300 400 500 600N

ββββ

0

2

4

6

8

10

12

14

λλλλaddβ = 3.54, V(β)=0.05, N=385 #LSFE=153, pf = 2.0 10-4

95% CI(β)

λadd=var

λadd=3.0

βT = 3.7

Figuur 17: DARS (λadd=var) - β en λadd in functie van het aantal simulaties N

GT procedure β pf #LSFE(tijd) nauwkeurigheid N

UGT(Atena2D) DARS λadd=var 3.54 2 10-4 153 (28h) V(β)=0.05 385

GGT (Calfem) 2.46 7 10-3 372 σ(β)=0.15 743

UGT (EC6) 1.60 5.5 10-2 78 σ(β)=0.15 243Tabel 17: Samenvatting betrouwbaarheidsanalyse in uiterste grenstoestand (UGT) engebruiksgrens toestand (GGT)

Een lineair-elastisch model kan wel gebruikt worden ter controle van de gebruiksgrenstoestand,waar het moment van optreden van de eerste scheur bepalend is. In dit geval wordt het eindigeelementen pakket Calfem gebruikt, dat ook in Matlab is geïmplementeerd. Zo kan een efficiëntecommunicatie tussen de betrouwbaarheidsprocedure en het eindige elementenmodel op puntworden gesteld. De verkregen betrouwbaarheidsindex beantwoordt aan de doelwaardenvooropgesteld in EC1: β=2.46>2.1. Annex A.15 geeft een tweede voorbeeld van de efficiëntewerking van de procedure wanneer een automatische koppeling kan worden voorzien tussen hetbetrouwbaarheidsalgoritme en de grenstoestandsevaluaties via eindige elementen evaluatie.

Tot slot wordt de veiligheid gecontroleerd overeenkomstig de procedure voorgesteld in EC6.Deze procedure is eerder elementair. De resulterende betrouwbaarheidsindex, Tabel 17, is laag(β=1.6) en stemt niet overeen met de waarde gevonden via de niet-lineaire eindige elementenbenadering.

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7. Besluiten en verder onderzoek

7.1. BesluitenEen algemene methodologie werd uitgewerkt om op een objectieve manier de veiligheid van eenbestaande structuur te beoordelen. Een niveau III methode of probabilistische methode wordtnaar voor geschoven. Daarin kunnen alle variabelen als toevalsvariabelen worden gedefinieerd,een vertaling van de aanwezige onzekerheid. Mogelijke doelwaarden voor de beoogdeveiligheidsniveau’s worden aangereikt.

Om het hoofd te bieden aan problemen met een groot aantal toevalsvariabelen en complexegrenstoestandsfuncties, worden de traditionele methodes gecombineerd met een aanpasbaarresponsoppervlak. De resulterende procedure zorgt ervoor dat het aantal rechtstreeksegrenstoestandsevaluaties tot een minimum wordt beperkt. De bestaande DARS procedure wordtdaartoe verder geoptimaliseerd. Een gelijkaardige procedure op basis van Monte Carlo simulatie,MCARS+VI, wordt uitgewerkt. De efficiëntie wordt gekwantificeerd aan de hand van 15academische voorbeelden, Appendix A.

De rekenmodellen die vereist zijn om het veilige van het onveilige gebied te onderscheiden,worden kritisch beoordeeld. De waarschijnlijkheidsverdelingen voor de belangrijkstemateriaaleigenschappen die in deze modellen voorkomen, worden bepaald. Waar mogelijkgebeurt dit experimenteel op het niveau van het metselwerk. Numerieke modellen voor dedruksterkte en elasticiteitsmodulus worden geverifieerd op hun bruikbaarheid om dewaarschijnlijkheidsverdelingen van het composiet te leveren op basis van de componentwaarden.Literatuurwaarden vullen dit experimenteel onderzoek aan. Daarnaast wordt ook hetmetselwerkgedrag onder multi-axiale spanningstoestand bestudeerd. Een sterktetoename enverhoogd plastisch gedrag worden vastgesteld, reeds bij lage steundrukken. De individueleexperimentele waarden en hun statistische verwerking worden toegevoegd in Appendix B.

Vier toepassingen tonen de bruikbaarheid van de methodologie aan voor ongewapendmetselwerk. Deze bundelen de opgestelde betrouwbaarheidsmethodes, dewaarschijnlijkheidsverdelingen voor de materiaaleigenschappen en de verschillenderekenmodellen die werden behandeld. Niet enkel het huidige veiligheidsniveau wordt op eenobjectieve manier berekend, maar ook wordt een aanzet gegeven naar de bruikbaarheid van detechniek bij de verdere opties in het conserveringsproces.

7.2. Verder onderzoekTot slot worden enkele pistes voor mogelijk verder onderzoek aangereikt:• in plaats van gebruik te maken van een 2de orde veeltermbenadering in het

responsoppervlak, verdient het de voorkeur op zoek te gaan naar een ‘universelebenaderingsfunctie’. Neurale netwerken hebben de eigenschap geen functionele vormvoorop te stellen, zeer flexibel te zijn en eender welk type functioneel verband tebenaderen.

• Een grafische gebruikersinterface verbetert ongetwijfeld de gebruiksvriendelijkheid.• Een koppeling tussen de betrouwbaarheidsprocedure en een eindig elementenprogramma

dat tevens geïmplementeerd is in Matlab heeft sterke troeven. Op deze manier kanontworpen worden volgens een vooropgesteld veiligheidsniveau.

• In dit werk werd de methode voornamelijk geïllustreerd op ongewapend metselwerk.

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Uiteraard kan dezelfde technologie gebruikt worden voor andere materiaaltypes.• Naast de evaluatie van het huidige veiligheidsniveau, dient meer aandacht te worden

besteed aan de toekomstige evolutie van het veiligheidsniveau bij vooropgesteldeingrepen en bij het afwegen van verschillende opties. Daar kunnen kosten/batenelementen aan worden toegevoegd (niveau IV methode).

8. Appendix A

Deze bijlage bevat 15 uiteenlopende voorbeelden waarin de betrouwbaarheid wordt berekend metverschillende technieken. De eerste 14 werden overgenomen uit de literatuur. Aanvullendworden de resultaten berekend met DARS (λadd=var) en MCARS+VI (λadd=3.0 en ∆g, add=var).De efficiëntie met betrekking tot volgende criteria wordt beoordeeld: meerdere kritieke punten(1), ruis (2), unies en doorsneden (3), dimensie van het probleem (n) (4), niveau van de faalkans(5), sterke krommingen (6) in de grenstoestandsfunctie of geen nulpunten volgens de hoofdassen(7). De resultaten zijn samengevat in Tabel 18.

nLSFE voorbeeld criteria DARS(λadd=3.0)

DARS(λadd=var)

MCARS+VI(λadd=3.0)

MCARS+VI(∆g,add = var)

1 Standaard R-S probleem / 18 9 41 5

2 LSF met ruis 2 271 41 246 31

3 R-S probleem + kwadratische term 4 38 13 32 9

4 LSF met 10 kwadratische termen 4,6 221 40 36 36

5 LSF met 25 kwadratische termen 4,6 188 86 86 82

6 Convex faaldomein 7 47 16 18 13

7 Scheve sfeer 1,4,5 160 65 762 61

8 Zadeloppervlak 3,6,7 225 19 46 15

9 Discontinue LSF 3 55 29 11 39

10 LSF met twee takken 3 135 138 12 92

11 Concaaf faaldomein 2,3,4,6 240 85 558 18

12 Serieel systeem met 4 takken 1,3 175 85 224 169

13 Parallel systeem met 4 takken 1,3 127 314 314 314

14 Aangepaste versie van LSF 13 1,3 51 51 79 192Tabel 18: Samenvatting van de efficiëntie bij de onderlinge vergelijking van verschillende procedures in combinatiemet een aanpasbaar responsoppervlak

Het 15de voorbeeld is de analyse van een vlak raamwerk. Daarin wordt de link gelegd tussen heteindige elementen programma Calfem en de betrouwbaarheidsprocedure. Omdat beide in deMatlab omgeving zijn geïmplementeerd, verloopt de onderlinge communicatie zeer efficiënt.Dit resulteert in een stochastische eindige elementen benadering (SFEM), waarbij het ontwerpkan gebeuren op basis van een vooropgesteld veiligheidsniveau.

-lii-

9. Appendix B

Deze bijlage bevat de individuele proefresultaten die zijn uitgevoerd in het kader van hetexperimenteel onderzoek, Hoofdstuk 5. Daarmee wordt niet enkel transparantie beoogt voor desamenvattende tabellen zoals ze in het werk zijn opgenomen. Het moet onderzoekers ooktoelaten gebruik te maken van de individuele data op zich.

Voor de uitgevoerde proeven zijn, waar opgemeten, volgende data voorhanden:• individuele resultaten voor elk proefmonster,• spanning-rek relatie,• geometrie,• histogram van de experimentele data met toegevoegd de geschatte

waarschijnlijkheidsverdeling,• quantiel plots (normaal en lognormaal),• breukpatroon,• locatie LVDT’s.

1

1 Introduction

1.1 General Framework

Safety, reliability and risk are key issues in the preservation of the built cultural heritage.Structural collapses such as the Basilica of San Franscis (Italy, 1997), the Chichester Cathedralspire (United Kingdom, 1861) (Buttress et al., 2001) and the Pavia Tower (Italy, 1989) (Binda,2000), make us aware of the vulnerability of our technical and natural environment and demandan adequate engineering response.

As it is a major concern of engineers, the notions of reliability and risk and related theory andpractice are no longer working topics for specialists only.

Safety and serviceability requirements for limit states design in North America and Europe havebeen developed and implemented in codes of practice during the past 20 years. Limit statecriteria in design codes are based on probabilistic methods. Until nowadays, the focus has beenon the design of new structures. However, the importance of assessing existing structures isgaining field (Frangopol and Kong, 2001; Shirariashi and Furuta, 1998; Vrouwenvelder, 1993).Guidelines for existing structures exist in several countries (USA, Canada, Switzerland, UK).Most of these guidelines refer to specific structures such as bridges or towers and containrecommendations associated with particular hazards such as seismic action. At present, only afew countries have a generally applicable and realistic code type for the assessment of existingstructures (CSN 73 0038, 1987 in Chechoslovakia; RBCV, 1992 in the Netherlands, SIA 462,1994 in Switzerland).

The Joint Committee of Structural Safety (liaison committee between different internationalorganizations, including: CIB, ECCS, FIB, IABSE, RILEM) undertakes efforts to harmonizebuilding codes at an international level in order to achieve a uniform structural safety approach.Besides the development of a probabilistic model code, extension to assess the reliability ofexisting structures is one of the main objectives (Diamantidis, 1999, 2001).

1.1.1 The need for assessmentWith approximately 10 000 listed and protected monuments, the Belgian Building Heritage isexceptionally rich. The preservation of this unique patrimony requires fundamental effort. TheKoning Boudewijn Stichting (Rosiers et al., 1998) estimated the total cost necessary to maintainor improve the building heritage to such an extent that it would only require regular maintenancefor a reference period of 50 years, to be 2.25 billion EURO.

Reliability-based design focuses on selecting and optimizing a structure (geometry, structurallayout, choice of material) which meets target service life reliability levels with respect tospecific limit states. On the other hand, reliability-based assessment deals with estimating theactual reliability level for a certain reference period based on all the information that can possiblybe gathered from the existing structure.

An assessment of an existing building can be called for in the following situations (Allen, 1991a;

2

Diamantidis, 1999, 2001; Ellingwood, 1996; ISO13822, 1997; ISO2394, 1998; Maes et al., 1999;Melchers, 1999):• an anticipated change of use, rehabilitation;• expiry of the residual service life granted on the bases of an earlier assessment of the

structure or extension of design service life;• a reliability check targeting specific hazards (e.g. wind loading, floor loads, earthquakes)

as required by regulatory agencies, insurance companies or owners; • observed or suspected deterioration related to building material, construction method or

the statical system due to time dependent actions (e.g. normal operating orenvironmental: frost damage, salts or pollution, fatigue, creep);

• structural damage caused by accidental actions (windstorms, severe fires, avalanches,earthquakes, impact of vehicles) leading to doubt about the structural safety;

• reliability updating following repairs or inspection results;• discovery of a design/construction error after the building is occupied;• complaints from tenants or operators regarding a clearly inadequate serviceability;• doubts about the safety of the structure.

The assessment of an existing structure may differ very much from the design of a new structure.Uncertainties in loads and resistance at the design stage are reflected in load and resistancefactors. At the evaluation stage, uncertainties in loads and resistances can be either greater thanat the design stage (e.g. hidden component or details, deterioration) or less (properties measured,load tests or satisfactory past performance). The range is therefore broader and the incentive tofind out what exists should be high (Allen, 1991b). Structural monitoring similarly provides afeedback which can lead to an update of the uncertainties. The information state is thereforecompletely different from that during design. However, the interpretation and the analysis of theadditional information may not be a simple matter.

1.1.2 The objectives of a reliability assessmentThe objective of assessing an existing structure in terms of its required future performance canbe defined in accordance with the following performance requirements:• life safety performance requirements, which provides minimum life safety for the users

of the structure (ISO13822, 1997);• continued function performance requirements, which provides for the continued function

of the structure subject to future use, including the possible occurrence of certain hazards;• special performance requirements, specified by the client, the owner or the regulator.

As an example of the latter, consider certain buildings such as hospitals which must also remainfunctional after a disaster such as an earthquake, and, in addition to life safety, damage controlis an added fundamental requirement for these buildings. Finally, additional structural protectionagainst damage beyond that required for life safety should be provided for specific failures if thereduction in expected loss (including heritage) due to future damage is justified by renovationcosts. In case of an historical building, one may wish to preserve, or enhance the value ofdifferent aspects of the structure. For instance, the original use of the materials, the authenticbuilding concept or the architectural history. In case one values the preservation of existingmaterials, the basic principle applies: only intervene when absolutely necessary. The "heritage"criterion amounts therefore to minimum destruction of existing materials and systems, either asa result of the renovation process or as a result of expected future building damage (Allen,

3

1991c).

1.1.3. The assessment frameworkThe architectural preservation process is generally based on a sequence of anamnesis andanalysis, diagnosis, therapy, control and prognosis (Croci, 1998; ICOMOS, 2001; Lemaire andVan Balen, 1988; Van Balen, 1997).

In the anamnesis and analysis phase, an objective way to assess the safety of the structure isessential. The present raises the need for a reliability based assessment framework for existingmasonry structures (TV, 2001; Diamantidis, 1999, 2001), Figure 1.1. It is in the detailed analysisphase, that a reliability based assessment fits into the framework. It is meant as an objectivemanner to assess the safety of the existing building, taking into account all kinds of uncertaintiesinherent to the structures’ state. In that it is an objective tool in the decision process. Theobjective in the prognosis phase is similar. It is meant as a tool to compare possible restorationoptions, to identify critical parameters and to derive an optimal solution, again accounting for allkinds of (future) uncertainties, such as future loading.

1.1.4 Different levels to assess a structures’ safetyNowadays, powerful methods are available for the calculation of structural safety values. Thesepermit to calculate the global probability of failure of complex structures, relying ondeterministic techniques able to determine the stability state for a prescribed set of parameters.

Level Definition

Level III Level III methods such as Monte Carlo (MC) sampling and NumericalIntegration (NI) are considered most accurate. They compute the exactprobability of failure of the whole structural system, or structural elements,using the exact probability density function of all random variables.

Level II Level II methods such as FORM and SORM compute the probability offailure by means of an idealization of the limit state function where theprobability density functions of all random variables are approximated byequivalent normal distribution functions.

Level I Level I methods verify whether or not the reliability of the structure issufficient instead of computing the probability of failure explicitly. In practicethis is often carried out by means of partial safety factors.

References: Eurocode EC1, Annex A; Joint Committee of Structural Safety 1982Table 1.1: Different levels for the calculation of structural safety values (EC1, 1991; JCSS,1982)

4

Visual inspectionPreliminaryevaluationStudy of actions tobe taken

Doubtaboutsafety

Phase 1

Phase 2

Phase 3

Responsiblestructuralengineer alone

CauseChange of use, Routineinspection, Degradation,…

Responsiblestructuralengineer aloneDetailed investigation,

Site investigation,Detailed structuralanalysisReliability analyis

Responsibleengineer with ateam of experts

Safetysufficient

Largeconse-quences

yes

no yes

no

yes no

No action taken

Intensify inspections

Restrict use of structure Strengthen Demolish, build new

Research institute-specialists

Inspect structureConfer togetherand decide as a team

Ana

mne

sis a

nd A

naly

sis

Dia

gnos

isTh

erap

y

Continuous assessment,reliability analysis

Reliability based design

Prog

nosi

s

Responsible structuralengineer alone

Phase 4a Phase 4b

Responsibleengineer with ateam of expertsResearch institute-

specialists

Figure 1.1: Global framework for the assessment of existing structures [adoptedfrom (Diamantidis, 1999,2001)]

5

The Joint Committee of Structural Safety, defines four levels at which the structural safety canbe assessed (JCSS 1982, Melchers, 1999), Table 1.1. Level IV adds economical data to the levelI, II or III methods to obtain maximum benefits and minimum costs. This is beyond the scopeof this study. The other three levels were adopted in the European Standard Eurocode 1 (EC1,1991). Level III methods are the most accurate. Level I and level II methods are simplifiedapproaches introduced for computational reasons. Ideally, they should be calibrated using a levelIII method.

1.1.5 Target safety for existing (historical) buildingsFor the decision process that follows up on a safety assessment of an existing structure, it isimportant to define target safety levels which can serve as decision criteria. In relation tofulfilling such criteria, existing (historical) structures will differ from new structures at the designstage (Arnjberg-Nielsen and Diamantidis, 1999; Diamantidis, 2001).

At the design stage, it relatively costs very little to provide a degree of structural safety thatexperience shows is very high. In other words, very little is gained in shaving safety factors inspecific situations in order to save money and this means that the generic criteria contained indesign codes, which cover all situations conservatively, are suitable for practice (Diamantidis,1999; Ellingwood, 1996). For structural evaluation however, the difference in cost betweenmeeting or not meeting a criterion, in other words the cost of upgrading, if required, can be large.The economics of upgrading therefore puts much greater pressure to determine specific criteriafor each situation, based on the fundamental requirements. Moreover, many current requirementsspecify quantities and arrangements of materials which are economical and practical toimplement during initial construction but impractical after a structure is completed (Maes et al,1999).

This is even more emphasized in case of historical buildings, were all interventions should meetthe Charter of Venice (Charter of Venice, 1964) or the Charter of Krakow (Charter of Krakow,2000). These imply extra implications (and costs) in case an intervention is required. Therefore,extra stress is put on the question whether or not an intervention is required. And if so, whatintervention will have the most benefit with respect to the structures’ safety, which of courseremains of primarily importance.

Specific efforts have been made to define acceptable failure probabilities for existing structuresand especially for offshore structures (Bea, 1993). The analysis of several platforms around theworld has shown a differentiation between new designs and requalification of existing platforms.This differentiation reflects a willingness to accept lower reliabilities associated with oldersystems and not to require that those systems have reliabilities that equal those of new systems.Such a willingness can be demonstrated to be true for a variety of engineering systems such ascars or offshore platforms (Diamantidis, 1999). There is no question that the same holds forstructures, in particular historical buildings.

Moreover, the residual service life is a key issue in determining target safety values. Often anhistorical building exists for a much longer period than its anticipated service life, although thelatter might never have been stated explicitly. As a building becomes protected or listed, a newlevel of performance may be imposed by the preservation authority. The new service life -

6

although not explicitly mentioned - might exceed the design reference life specified in moderndesign codes.

As the models (structural models, load models, material models) are only partly based onexperimental data, the calculated failure probability should not be identified directly with actualfailure frequencies (Holicky and Vrouwenvelder, T2881,1999). Therefore, they should bereferred to as nominal failure probabilities. They are intended to obtain objective criteria to judgethe structural safety.

1.1.5.1 Target failure probabilityDetermining the target probability of failure is not a technical problem solely. Whether or notan historical building should meet the target probability of failure value pfT = 5.10-4 accordingto the Belgian Standard NBN B03-001 (1988), is subject of discussion. Several authors suggestto widen the discussion (Allen, 1982; Ditlevsen, 1982; EC1, 1994; IS2394, 1998; Melchers,1999; NEN 6700, 1991; Van Dyck, 1995) and propose a differentiation with respect to variousperformance criteria discussed above and some additional parameters:• the type of possible damage (life injury, economical damage, social-cultural damage,

environmental damage),• the preset service life of the construction,• the level in which people are exposed to risk (public buildings, bridges, off-shore

constructions, ...) and• the level in which people are warned beforehand and can be put to safety (gradual failure

with visible damage against sudden collapse without warning).

Empirical formulas are available to calculate a nominal value for the target probability of failurepfT (Melchers 1999; Van Dyck 1995). The Construction Industry Research and InformationAssociation (CIRIA, 1977), proposes to determine the probability of failure for a preset designservice live (tL) following:

(1.1)p S tnfT c

L

p

= −10 4

in which np is the average number of persons in the immediate neighborhood of the building orin the building itself. The factor Sc transcribes a social criterion. Their values are listed in Table1.2.

A second empirical formula is developed by Allen (Allen, 1991b):

(1.2)p tn

AWfT

L

p

c= −10 5

In this formula, Ac is an “activity” factor, W is a “Warning” factor. These values are listed inTable 1.2. The "activity" factor reflects what risk is acceptable in relation to other non-structuralhazards associated with the activity, the "warning" factor corresponds to the likelihood that, givenfailure or recognition of approaching failure, a person at risk will be killed or seriously injured.

7

Both proposals account for the risk of the number of people with a different weight. Besides,none of both account for the number of injuries, nor with the economical cost of failure. Theycan serve as indicative values. Some standard codes specify explicitly the target probability offailure (CEB, 1976, 1978). This effect can easily be fit in to the former equations by adding acost factor Cf, table 1.2.

The first formula, Eq.1.1, is very suitable, as it accounts for a social criterion, that can be re-interpreted to encapsulate the importance of historical buildings or the preservation value. Thesecond formula, Eq. 1.2, allows for more differentiation. Adding a cost factor and re-assigningthe social factor, following formula is proposed to determine nominal target failure probabilitiesfor historical buildings (added between brackets in Table 1.2):

(1.3)p S tn

AW

CfT cL

p

cf= −10 4

Economical factor Cf Warning factor W

not seriousseriousvery serious

1010.1

- Fail-Safe condition- Gradual failure with some warning likely- Gradual failure hidden from view- Sudden failure without previous warning

0.010.10.31.0

Activity factor Ac Social criterion factor Sc

- Post-disaster activity- Normal activities : Buildings Bridges- High exposurestructures (offshore)

0.3

1.03.010.0

- Places of public assembly, dams (historicalbuildings of great importance for mankind,listed by UNESCO e.g.)- Domestic buildings, offices, trade buildings,industrial buildings (listed historicalbuildings)- Bridges- Towers, masts, off-shore structures

0.005

0.05

0.55

Legend: social criterion factor Sc (CIRIA, 1977); “Activity-” (Ac) and “Warning” (W)factor (Allen, 1991b); impact of economics (Cf) according to (CEB, 1976, 1978)

Table 1.2 : Factors influencing the nominal target failure probability

8

1.1.5.2. Target reliability indexCodes that refer to calculations on level II and III, base their target values on the target reliabilityindex βT to measure the safety. This index is related to the target failure probability (pfT) bymeans of the standard cumulative distribution function Φ:

(1.4)( )pf = −Φ β

Their relation is listed in Table 1.3. In subsequent tables, both values will be listed for ease ofcomparison.

pf 10-1 10-2 10-3 10-4 10-5 10-6 10-5

β 1.3 2.3 3.1 3.7 4.2 4.7 5.2Table 1.3: Relation between pf and β

The target reliability levels proposed by various national and international associations are valideither for lifetime, either for a given reference period of typical one year. The relationshipbetween the reliability index associated with two reference periods t2 (lifetime) and t1 (one year),can be approximated in accordance with Eq 1.1-1.3, by:

(1.5)( ) ( )Φ Φ− = −β βT Ttt2 1

2

1

When national and international codes are reviewed (EC1, 1994; ISO 2394, 1998; ISO 12833,1997; NBCC-part 4, 1990; NEN6700, 1997), the listed target reliability values fluctuate, Table1.4. As recent designs are calculated based on the limit state concept and the partial safetyformat, both ultimate limit states (ULS) and serviceability limit states (SLS) are referred to.

Limit State Target reliability index βT (design working life)

Ultimate limit state 3.8(1)[7 10-5], (3.1;3.8;4.3)(2),(3)[10-3;7 10-5;9 10-6],3.6(4)[2 10-4], 3.5(5)[3 10-4]

Serviceability limit state:- reversible- irreversible

(0.0)(1)[0.5]1.5(1),(2),(3)[7 10-2], 1.8(4)[4 10-2]

Legend: (1)EC1, Annexe A, 1991 (2)ISO 2394, 1998; (3)ISO 12833, 1997; (5)NEN 6700,1997; (6)NBCC-part 4, 1990

Table 1.4: Target reliability values βT [pfT] according to International standards

Final tentative target reliability values are listed in Table 1.5 (Diamantidis, 1999,2001). Besideslife risk, the differentiation towards consequences of failure, allows for some interpretationtowards preservation values of historical buildings for instance.

9

Relative costsof safetymeasures

SLS(irreversible)

ULS - Consequences of failure

some moderate great

high 1.0 [0.2] 2.8 [3 10-3] 3.3 [5 10-4] 3.8 [7 10-5]

moderate 1.5 [7 10-2] 3.3 [5 10-4] 3.8 [7 10-5] 4.3 [8 10-6]

low 2.0 [2 10-2] 3.8 [7 10-5] 4.3 [8 10-6] 4.8 [8 10-7]Table 1.5: Tentative target reliability values βT [pfT](Diamantidis, 1999, 2001)

1.2. Building Materials and Building Technology Division

This doctoral research fits into a global restoration research project that is a main research fieldof the Building Materials and Building Technology Division of the Department of CivilEngineering of KULeuven. This research enables the institute to act as a specialized researchteam in the general assessment framework as shown in Figure 1.1. Indeed, different aspects ofthe assessment framework are research topics of different researchers that are or have beenworking at the Building Material and Building Technology Division.

As for the material behavior and material properties of masonry, research is conducted in thefield of historical (lime) mortars (Hayen et al., 2001a; Van Balen, 1991) and the behavior ofmasonry in triaxial stress state (Hayen, 1999; Schueremans et al., 2001b). As part of repair andstrengthening technique, the rheological, physico-chemical and mechanical properties of limebased multi-blend grouts are studied (Toumbakari, 2001).

In the analysis phase, different test techniques are available to gather information about thequality of the masonry. Seen the increased emphasis on non-destructive test methods, focus ison geo-electrical measurements (Venderickx, 2000), providing information about the internalstate of the masonry. This non-destructive test method is useful not only in the initial analysisphase, but is also a convenient way to judge the executed consolidation by means of groutinjection.

With respect to the structural behavior of masonry (Schueremans, 2000), the study of masonryarches and vaults (Smars, 2000) has led to a user-friendly software application Calipous. Thislimit analysis program calculates thrust lines and safety factors of arches using non-tensionmaterial behavior for masonry. Its use will be illustrated in this research work.

For consolidation, strengthening and repair of masonry, different techniques are available,amongst which the technique of grout-injection. The influencing parameters such as mechanicalproperties, stability of grout, bleeding, penetrability and fluidity have been studied in detail (VanRickstal, 2000).

Of course, masonry is not the only material encountered in restoration practice. Reinforcedconcrete structures may suffer from degradation for example caused by chloride attack(Poupeleer, 2001; Schueremans et al., 1999). Recent techniques focus on external strengthening

10

of reinforced concrete structures using carbon fibre reinforced plastics (CFRP) (Ahmed, 2000;Brosens, 2001). Similar techniques can be used for wooden structures (Desmidt, 1990;Horckmans, 1988; Ignoul et al., 2001; Schueremans et al., 2001b; Taveirne, 1989;).

1.3. Scope of this thesis

The scope of this thesis is twofold. On the one hand, a general methodology is looked for toobtain objective safety values for existing structures, on the other hand, the methodology focuseson unreinforced historical masonry structures.

The safety value, failure probability or reliability index of a whole system or structure or astructural element is aimed at. Calculating the failure probability means calculating theprobability that a limit state function is exceeded. In some cases the limit state function is knownanalytically, in other cases each limit state function evaluation will require a (non-linear) finiteelement calculation. The (ultimate) limit state functions for masonry and calculation models(both analytical and finite element models) are retrieved and rephrased for assessment purposeswhere necessary.

Different commercial generic finite element codes are available that can be used for modelingthe masonry material. Their applicability in combination with a reliability method will bediscussed. Since finite element computations take considerable time, the number of limit statefunction evaluations should be minimized.

To obtain objective safety values, a level III method (probabilistic approach) will be used.Therefore, it should be possible to treat all variables as random variables, accounting for thepresent uncertainties. These include:- material properties (strength and stiffness properties). The material properties that govern themasonry material behavior will be dealt with in detail,- geometrical properties (cross-sections, eccentricities). This allows to account for uncertaintiesresulting from a measurement survey.- loads (forces and displacements such as settlements). These will be accounted for, but are nopart of the research work at hand.

A large number of monuments and buildings are made out of brick masonry. This study mainlydeals with (historical) unreinforced masonry buildings representing the cultural/architecturalheritage of each country. The historical aspect puts extra complexity towards obtaining(experimental) data for the material properties.

Dealing with masonry structures of course only affects the material model and structural models.The general methodology should be applicable to other building materials as well when sufficientinformation concerning the material and structural model is available.

1.4. Outline of this thesis

11

In Chapter 2, a comprehensive overview is given of different reliability methods that areavailable for calculating safety values. A state of the art of available reliability methods is given.Their capability in obtaining global failure probability values according to a level III approachis evaluated.

The emphasis however is on newly developed combined methods. These methods combinetraditional reliability procedures such as Directional Sampling, Monte Carlo Sampling, with anAdaptive Response Surface.

In Chapter three, a recently developed method, Directional Adaptive Response surface Sampling(DARS), is described in detail. Improvements are added to the DARS procedure, increasing itsoverall efficiency. Also a promising alternative is proposed, MCARS+VI, in which Monte CarloVariance Increase is combined with an Adaptive Response Surface. Although generality ispursued, some specific comments towards masonry structures are already made. To illustratethese methods on a variety of limit state functions of varying complexity, 15 academic examplesare added, Annex A.

Chapter four introduces the different calculation models that are available for masonry structures.These are required to distinguish the safe from the unsafe region. In several cases the borderbetween safe and unsafe, can be expressed analytically, in other cases more complex numericalmodels are required. Historical masonry is mainly designed to act in compression. Analyticalmodels dealing with eccentric vertical loading are treated in detail. Reference is made to theiruse in current Standards. Compression along the shape of the masonry structure, arching, resultsin a limit analysis. Often however, finite element analyses are required to obtain a sufficientsimilarity with reality. A global description of numerical models for masonry is given. Availablecommercial (generic) finite element programs are checked on their applicability for masonrymodeling. These will be used in combination with the reliability methods outlined in chapter twoand three.

Chapter five summarizes the experimental research that was conducted in the framework of thisthesis. Major goal was to provide the required numerical data to feed the material models. Focusis on the different sources that are available to gather experimental data, different non-, semi- ordestructive tests and the statistical processing of data. The particular behavior of masonry undertriaxial circumstances is treated, for which a unique testing device is developed. The individualtest results can be consulted in Annex B.

Chapter six illustrates the applicability of the methodology. Several applications of varyingcomplexity are given to demonstrate the effectiveness and efficiency of the method: efficiencyof grout injection (analytical model), eccentric loaded column (analytical model), safety ofmasonry arches (limit analysis) and masonry shear wall (finite element analysis). Not onlyanalytical limit state functions are treated, but also non-linear finite element methods.

Final remarks, conclusions and suggestions for further research, are given in Chapter seven.

12

13

2 Reliability analysis - literature review

2.1 Introduction

This chapter is the first of two chapters that deal with the calculation of the reliability ofstructural systems. Although this branch of science is relatively young, different methods havebeen developed and optimized. The objective of this chapter is to review existing methods andplace these methods in a global reliability framework, pointing out their strong sides anddrawbacks. Several methods are used in the structural applications, Chapter 6. The subsequentChapter will focus on new developments in reliability procedures, combining traditional methodssuch as sampling procedures with an adaptive response surface.

First, the generalized reliability problem is outlined, Section 2.2. The required accuracy iscommented in Section 2.3, and different reliability methods are presented. Numerical as well asanalytical integration methods are focused on, Section 2.4. Sampling procedures such as MonteCarlo sampling and directional sampling are treated in Section 2.5. Finally, first order andsecond order reliability methods are reviewed, Section 2.6.

Their applicability, limitations and accuracy with respect to the evaluation of existing masonrystructures are discussed. For ease of comparison, the different methods are illustrated using thesame mathematical example. Additional examples are provided in Annex A.

2.2 Formulation of the generalized reliability problem

The generalized reliability problem can be outlined starting from the basic reliability problem(Melchers, 1999; Van Dyck, 1995). The basic reliability problem considers only one load (S1)and one resistance (R1). Starting point is that both the load effect (S1) and the resistance (R1) arerandom variables. Each is described by a known probability density function: fS1(s1) and fR1(r1).In general, both the load effect (S1) and the resistance (R1) are a function of time t. Loads (S1)tend to increase, resistances (R1) tend to decrease due to all kinds of degradation processes. Thesafety limit state will be violated when at a certain moment in time, R1(t) - S1(t) < 0, Figure 2.1.The chance or probability that this will happen equals the failure probability pf. As both R1 andS1 are a function of time, pf is also a function of time.

Initially, this chance will be very limited in normal practice. For many historical masonrystructures, due to the high initial safety, the safety will remain very high, even at the commondesign service life (t1 = 50 years). When degradation processes start to influence the strength(R1), the failure probability starts to increase. At a certain moment in time, t2, an assessment iscalled for. When an increased and unacceptable failure probability is stated, an interventionbecomes inevitable.

A wide scatter of intervention possibilities is available. In case the expected load (S1) is notincreased significantly, one may decide not to intervene in the structure's authenticity andrehabilitate where necessary without strengthening (R1). When the failure probability is too

14

fR1(r1|t=t1)

fS1(s1|t=t1)

fR1(r1|t=t2)

fS1(s1|t=t2)

R1(t)

S1(t)S1(t)

S2(t)

R3(t)

R2(t)

pf(t)

Figure 2.1: Generalized R-S-problem

excessive, one may decide to strengthen the building, up to the original level (R3) or up to anincreased level (R2) when for example an increase in future load is expected (S2).

Because of mathematical complexity, the probability density functions are transformed into time-invariant probability density functions. As a consequence, the reliability analysis is performedfor a preset reference period or so-called design service life tL:

pf = P[ R1-S1 < 0], for a reference period tL. (2.1)

This step has major consequences on the further calculations. Load probability density functions(S) are translated into extreme value distributions such as Gumbel (EV-I) or Frechet (EV-II).This will not be dealt with in this study. Several references can be found on this subject, dealingwith distribution functions and their parameters for live loads in domestic buildings and offices(Chalk et al., 1980; Corotis and Doshi, 1977; Ellingwood and Culver, 1977; Harris et al., 1981;Peir and Cornell, 1973; McGuire and Cornell, 1974), wind loads (Simiu and Filiben, 1980; Simiuet al, 1978), bridge loads, snow and wind loads (Der kiureghian, 1980; Kasperski and Holmes,1999; NBS 577, 1978; NBS 110, 1978; Naes and Leira, 1999).

Estimating the time dependency in the distribution density function for the resistance (R) in caseof masonry is not evident at all. Masonry creep models contain several parameters, that canhardly be defined (De Raeymaecker, 2001). Ideally, the masonry resistance is not affected by anydegradation process. This means that its density function is time invariant, which would verymuch simplify the calculations. In reality however, different types of degradation processes mayaffect the resistance of masonry (Van Balen, 1998). These may lead to a gradual decrease in

15

resistance. If one would like to incorporate this effect into the resistance probability densityfunction, the effect of the degradation process on the resistance has to be quantified. As anexample, the effect of freeze-thaw cycling on the masonry resistance is reported elsewhere (Maes,Schueremans and Van Balen, 1999). When the degradation process can be described and theparameters estimated, a time dependent reliability analysis can be performed. As for otherbuilding materials, interest in this field is growing. For reinforced concrete structures, the effectof chloride ingress as degradation process is studied using probabilistic methods with increasingsuccess (Schueremans and Van Gemert, 1997a, 1999d). Relatively small research effort howeveris found in literature with respect to the subject of this thesis. No universal models norparameters could be retraced.

For the research at hand, the resistance (R) probability density function is determined on themoment an assessment is asked for, based on in site testing. These results also account for thedegree of deterioration at that moment in time. In case of restoration, further deterioration, ifpresent, will be prohibited using stone hardeners, water-repellent treatments, replacement ofbricks, grout-injections or others. By that, the influence of time on the resistance values (R2 orR3) will be limited, Figure 2.1. In the following, only time invariant reliability analyses areperformed (Schueremans et al., 1997b).

Equation (2.1) is generalized as follows:

pf = P[g(X) < 0], for tL, (2.2)

in which g(X) is called the limit state function, the vector X is a set of n random variables (n =2and X = [R,S] for the basic reliability problem) and the probability of failure is identical with theprobability of limit state violation.

This limit state function defines 3 different regions:

- g(X) > 0, the safe region,- g(X) = 0, the critical situation and (2.3)- g(X) < 0, the unsafe region.

For the most general case, the failure probability is defined as:

(2.4)( )[ ] ( )( )

p P g f dfg

= ≤ =≤

X x xXX

00

...

in which fX(x) is the joint probability distribution function of the n-dimensional vector X.

16

0

1

2

3

4

5

6

1 2 3 4 5reliability index β

β

1,E-08

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

V(β) or V(pf)σ(β) or σ(pf)combined criterion

β=

pf

Figure 2.2: Preset accuracy versus reliability index β

2.3 Required accuracy

Before discussing the different methods to solve the generalized reliability problem, someattention is paid to the accuracy required in these calculations with respect to the failureprobability. As will be outlined further on, a lot of effort is put in minimizing the amount of limitstate function evaluations (nLSFE) as they often require the greater part of computation time. This,of course, is only meaningful under the constraint of sufficient accuracy.

Waarts (Waarts, 2000) proposes to limit the coefficient of variation of the reliability index to 5%:V(β) = 0.05. This accuracy is mainly proposed for the design of new structures and is based ona target reliability βT = 3.8. In that case, the 95% confidence interval for the reliability indexequals: CI(βT)=[3.49, 4.11] and for the corresponding failure probability: CI(pf)=[2.0 10-5, 2.410-4], Figure 2.2. So, for practical use, the only interest in fact is a good estimate of the order ofmagnitude of the failure probability, because the coefficient of variation equals: V(pf) = 0.52.

In case of structural evaluation, the target accuracy should be similar. Therefore, an importantremark needs to be made. When an assessment is asked for, one doubts whether or not the targetreliability of 3.8 will be met. In some cases, the actual reliability (βa) might only be βa = 1.5.In that case, given the preset accuracy of V(β) = 0.05, the 95 % confidence interval for thereliability index becomes: CI(β)=[1.38, 1.62] and the corresponding failure probability:CI(pf)=[0.052, 0.084], which means only about 3 tenth of an order of magnitude (0.084 - 0.052= 0.032). This accuracy will cost more computational effort.

Therefore, in case of low reliability values (β=3.5 is considered as turning point), it may be moreconvenient to introduce an absolute criterion and to restrict the accuracy to a preset standarddeviation of σ(β)=0.15. Following criterion is proposed:

17

(2.5)( ) ( )σ β β β β= ≤ = ≥015 3 00 0 05 3 00. , . : . , .for and V for

In case β = 1.5, the 95 % confidence interval for the reliability index becomes: CI(β)=[1.25,1.75] or V(β) = 0.10 and the corresponding failure probability: CI(pf)=[0.04, 0.12], or:V(pf)=0.36.

From Figure 2.2 it can be seen that the accuracy on the reliability index or failure probability isspread more equally as a function of the present reliability index. Main consequence of thepreset accuracy is the computational effort. When the problem at hand is rather simple, abovementioned accuracy is not an issue. Any arbitrarily chosen accuracy can be reached. On theother hand, when external iterative non-linear finite element methods are involved, theconsequences on the computation time could be huge.

2.4 Overview of standard reliability methods

An overview of common reliability methods is given in Table 2.1. Besides the level of reliabilitymethod (I, II or III), it is indicated whether or not the limit state function is evaluated directly (D)or the reliability analysis is performed on an estimated response surface, so-called indirectevaluation (ID).

In the following sections, the different methods will be reviewed shortly. Some of these methodswill not be used individually, but in combination with other methods. In pointing out their strongsides, it becomes clear why certain methods are combined and why a certain method is used afteranother and not before it. It is not the authors’ goal to achieve completeness in description, onlyto provide enough information for practical use. For more information, the reader is referred tostandard work on this matter (Madsen et al., 1986; Melchers, 1999), or prominent authors in thisfield such as Breitung, Der Kiureghian, Ditlevsen, Madsen, Melchers and Rackwitz.

The different reliability methods are illustrated using a mathematical example, which is furtherelaborated in detail parallel with the theoretical explanation.

Some of the reliability methods solely work with standard normal independent random variables(u-space). Each set of n random variables (x-space) can be transformed into a set of m randomvariables (m @ n) in the independent standard normal space (u-space). The dependent randomvariables are first remodeled into standard normal variables by equating the cumulativedistribution functions for the basic random variable and the standard normal variable, furtherdenoted as the normal tail extension (Melchers, 1999):

(2.6)( ) ( ) ( )( )Φ Φu F x u F xi X i i X ii i= ⇔ = − 1

Reliability methods - Level (I, II, III) - Direct/Indirect (D,ID)

Integrationmethods

Analytical or Numerical integration (AI/NI, III, D)

18

Directional Integration (DI, III, D)

Samplingmethods

(Importance Sampling) Monte Carlo ((IS)MC, III, D)

(Importance) Directional Sampling ((I)DS, II, D)

FORM/SORMmethods

First Order Second Moment Method (FOSM, II, D)

First Order and Second Order Reliability Method (FORM/SORM)(level II) in combination with a system analysis (FORM/SORM-SA, III,D)

Combinedmethods usingAdaptiveResponseSurfaceTechniques

FORM/SORM with an adaptive Response Surface (ARS) (level II) incombination with a system analysis (III)(D-ID)

Directional Adaptive Response surface Sampling (DARS, III, D-ID)

Monte Carlo Adaptive Response surface Sampling (MCARS, III, D-ID)

Table 2.1: Overview of reliability methods for a level III reliability analysis

In case of correlated random variables several methods exist to transform these correlatedrandom variables into non-correlated random variables, such as the Rosenblatt transformation(Rosenblatt, 1952) and Nataf transformation (Johnson and Kotz, 1972; Nataf, 1962). Afrequently used transformation for normally distributed correlated variables can simply beperformed by calculating the eigenvalues and eigenvector of the matrix containing thecovariances (CX) and applying a modal decomposition (Stange, 1971, Melchers, 1999). Remarkthat there are n! possible transformations, depending on the arrangement of the eigenvectors. Incase of non-normal dependent random variables, each arrangement may result in a differentestimate of the failure probability (Dolinsky, 1983).

Example - serial system with 4 branchesFollowing limit state function will be used to illustrate the different reliability methods:

(2.7)( )

( ) ( ) ( )

( ) ( ) ( )

( )( )

g u u

g u u u uu u

g u u u uu u

g u u u ug u u u u

1 2

1 1 2 1 22 1 2

2 1 2 1 22 1 2

3 1 2 1 2

4 1 2 2 1

2 0 0 12

2 0 0 12

2 5 22 5 2

, min

, . .

, . .

, ., .

=

= + − −+

= + − ++

= − += − +�

The example already has been analyzed by different authors (Katsuki and Frangopol, 1998; Borri andSperanzini, 1997; Waarts, 2000). It is slightly adapted to obtain a lower reliability index, which is moreillustrative for assessment problems. This example is taken as it is sufficiently simple to illustrate themethodology and has enough complexity to deal with the advantages and drawbacks of the different

19

u1

u2

g1(u1,u2)<0

g2(u1,u2)<0

g3(u1,u2)<0

g4(u1,u2)<0

unsafe

unsafe

unsafe

unsafe

safeg3>0 g1>0

g4>0g2>0

Component reliability:

pf,g1= 0.0161, βg1=2.14pf,g2= 0.0161, βg2=2.14pf,g3= 0.0062, βg3=2.50pf,g4= 0.0062, βg4=2.50

System reliability:

pf= 0.0446β=1.70

Figure 2.3: Problem definition 2D - limit state function and contours of joined probability density function

methods. The method will be outlined in the standard normal space, u-space. This prohibits extratransformation to independent and standard normal variables. To allow graphical representation, theexample is limited to 2 standard normal random variables (u1, u2):

(2.8)u N u N1 20 1 0 1≈ ≈( , ); ( , )

The limit state function and probability density function of the random variables u are shown in Figure 2.3and Figure 2.4.

20

u1u2 g1(u1,u2)<0

g2(u1,u2)<0

g3(u1,u2)<0

g4(u1,u2)<0

unsafe

unsafeunsafe

unsafe

u2u1

fU1,U2(u1,u2)

fU1,U2(u1,u2)

g1(u1,u2)<0unsafesafe

Figure 2.4: Problem definition 3D - limit state function and joined probability density function

The component failure probabilities and reliability indices as well as the global failure probability (pf) andcorresponding reliability index (β) are indicated on Figure 2.3. With a preset accuracy in terms of standarddeviations σ(β) = 0.15, the 95% confidence interval for the failure probability and reliability index are:

(2.9)( ) [ ]( ) [ ]

95 0 026 0 07495 145 195

% . ; .

% . ; .

CI p

CIf =

=β

2.5 Analytical and numerical integration

Both analytical and numerical integration are level III methods and result in an exact value forthe global failure probability of a whole structure. Analytical integration of the generalizedreliability problem however is only possible for some very special cases (Madsen et al., 1986;Melchers, 1999). Therefore it is of limited practical interest. Numerical integration is apossibility that can be considered in case of a limited number of variables. Typically, thepractical limit for numerical integration is considered to be around n@5. The computationaldemands increase rapidly with the dimension n of the integration space (Melchers, 1999). Waarts(Waarts, 2000) estimated that in case β = 4 and a preset accuracy V(β)=0.05, the computationaleffort is proportional to 9n. Although computer time tends to decrease, numerical integration isapplied only for validation of other methods while using a low number of variables.

21

u1

u2

g2(u1,u2)<0

g3(u1,u2)<0

unsafe

unsafe

u’1

u’2

pS,1

safe

unsafe

g1(u1,u2)<0

g4(u1,u2)<0

unsafe

Pf = 1-4*pS,1=0.043β = 1.72#LSFE = 6

Figure 2.5: Analytical integration in the rotated u

Example Numerical integrationFor the numerical integration, no use is made of any a priori knowledge. The failure probability iscalculated according to Eq. 2.10:

(2.10)( )[ ] ( ) ( ) ( )p I g u u f u u u ufuu

≈ < = = == −= −

1 2 1 2 1 24

4

4

4

0 0 8 0 8 0 043621

, , . . .U ∆ ∆

The exact result will be obtained when the boundaries (u1=[-4,4], u2=[-4,4]) are extended to -Q and Q andwhen the integrators ∆u1 and ∆u2 are approximating zero. Using integrators ∆u1 and ∆u2 equal to 0.8 issufficient to obtain a global failure probability result that lies in between the preset confidence interval. Thenumber of limit state function evaluations, needed to evaluate the indicator function I[ ] amounts:

(2.11)# LSFE n= =9 81

which is equal to the value preset by Waarts (Waarts, 2000).

Analytical integrationAnalytical integration can only be performed when the limit state function is known analytically, which isassumed for this part of the example. Use is made of the symmetry across the first bisector. Therefore,the surface under the joined probability density function in the failure domain of only 1/4th needs to becalculated. Because of computational efficiency, use is made of the complement. In that, the safety

22

probability (pS=4*pS,1) is calculated instead of the failure probability:

(2.12)p p pf S s= − = −1 1 4 1* ,

Furthermore, the integration is not performed in the u-space, but in a u’-space that is rotated over 45degrees. This simplifies the analysis to a great extent and limits the number of direct limit state functionevaluations. The rotated axes, the area that is used for calculating the safety probability and a summaryof the results are shown in Figure 2.5.

The number of limit state function evaluations is very limited because of optimal use of prior knowledge.The LSFE are only necessary to calculate the integration boundaries.

2.6 (Importance Sampling) Monte Carlo

2.6.1 Crude Monte CarloThe Monte Carlo technique involves sampling at random of the random variables (X) from theirproper distribution fX(x) and calculating the relative number of simulations for which the limitstate is violated (g(x)<0). The probability of limit state violation as expressed in the generalreliability problem Eq. 2.2. , can be written as:

(2.13) ( )[ ] ( )p I g f df = ≤x x xX0

where denotes an indicator function which equals 1 in case the outcome is true. ( )[ ]I g x ≤ 0This is done to simulate artificially a large number of experiments and to observe� , � ,..., �x x x1 2 Nthe result. An unbiased estimate of the failure probability equals:

(2.14)[ ] ( )[ ]p E pN

I gf f ii

N

≈ = ≤=

� �1 0

1

x

Obviously, the required number of trials N is related to the desired accuracy for pf. The samplevariance Spf

2 is given by (Melchers, 1999):

(2.15)( )[ ] ( )SN

I g N pp ii

N

ff

2 2

1

211

0=−

≤ −�

��

�

��

=

� �x

The minimum required number of samples in Monte Carlo sampling, given a target coefficientof variation for the failure probability, V(pf), can be calculated (Vrouwenvelder, 1984; Mann etal., 1974):

23

(2.16)( )

NV p p

f f

≥ −�� ��1 1 12

Given a preset value V(pf) = 0.52, see section 2.3, this leads to approximately:

(2.17)Npf

≥ 35.

Given a 95% confidence level, α = 0.05, Broding suggests (Broding et al., 1964) that a firstestimate for the number N of simulations for a given confidence level α = 0.05 in the failureprobability pf can be obtained from :

(2.18)( )Np pf f

≥ =ln α 3

The difference in estimated number N required to reach a sufficient accuracy is small in Eq 2.17and Eq. 2.18 It is evident that the number of simulations increases drastically for low values ofsafety probabilities. Nevertheless, the number N is independent of the number of randomvariables n. Further, for higher failure probabilities, e.g. pf = 10-1, 10-2, the number of samplesmight remain acceptable. Finally, it should be stressed that this method is far most the simplestmethod to calculate the global failure probability on a level III basis. The exact probabilitydistribution of each random variable is used and system behavior is accounted for. Moreover,the limit state function must not be known explicitly, which makes it useable in combination withfinite element analysis. Each sample only requires a single limit state function evaluation (orfinite element calculation). Because of its simplicity, this technique is very fast in case the limitstate function has a simple functional form. Therefore, this technique is very attractive to use incombination with response surfaces, as often a polynomial of low order is used, Section 3.2.

2.6.2 Importance Sampling Monte CarloAs the required number of simulations is the main reason for not using the 'crude' Monte Carlomethod, so-called variance (Spf

2) reduction methods have been proposed (Melchers, 1999). Ascan be seen from Figure 2.6. the region of interest (gi()<0) is often very restricted, certainly inthe case of a single failure mode. Variance reduction can only be achieved by using additional(a priori) information about the problem to be solved. A number of techniques exist. In eachcase information about the problem is used to limit the simulations to interesting regions(Melchers, 1999; Maes et al., 1993). Therefore, this technique is called importance sampling.

The general reliability problem, Eq. 2.2. can be rewritten using an importance sampling functionhv(v):

(2.19)( )[ ] ( )( ) ( )p I g

fh

h df = ≤vvv

v vX

VV0

24

An unbiased estimate of pf is then given by:

(2.20)[ ] ( )[ ] ( )( )p E p

NI g

fhf f i

i

ii

N

≈ = ≤=

� ��

�

1 01

vvv

X

V

It is evident that hv(v) governs the distribution of the samples. How this distribution should bechosen is quite important. An appropriate distribution for hv(v) is considered to be (Engelundand Rackwitz, 1993):

(2.21)( ) ( ) [ ] [ ]h with E V andsymm

h h n

h

h n

V V Vv v C C= = = ���

���

Φ , , : :, ,

,

,

µ µσ

σ1

12

2

0� �

where CV is a strictly diagonal matrix of variances σh,i2.

A first possibility is denoted as importance sampling, where the mean v is placed at the failurepoint x*. Such a sample distribution will produce sample points unbiased with respect to eachvariable. It will also give greater sampling density in the neighborhood of the region where it isof most significance. Of course, the failure point or design point x* has to be known a priori.Therefore, importance sampling can only be performed after a method has been used which isable to determine this failure point, such as: FORM, SORM or DARS. Furthermore, for optimaluse, the failure point should be unique. If this is not the case, importance sampling still ispossible using more than one weighted sampling density function or a so-called multi-modalsampling function (Melchers, 1999). This will lead to higher complexity and loss of efficiency.Additionally, the mathematical gain depends on the form of the limit state function. In manypractical cases however, the limit state function is nearly linear or quadratic in a region aroundthe failure point. These are handled with ease. Extremely concave limit state functions howevermay lead to very low sampling efficiency. Because many factors may influence the efficiency,it is difficult to estimate the sampling efficiency numerically beforehand. Waarts estimates thatfor optimal choice of sampling function, β = 4 and V(β)=0.05, the number of samplesapproximately equals N=300n, thus depends on the number of random variables (Waarts, 2000).

A second possibility is denoted as variance increase. By increasing the variance, sampling pointsare generated in a wider area, increasing the number of samples in the failure area. The optimalvariance depends on the number of random variables. An experimental relation exists, based onsimulation results (Waarts, 2000):

(2.22)σ βh n= −0 4.

Again, a priori information, in this case an estimate of the reliability index β, is required.When prior knowledge is not available, importance sampling may not be efficient. As thesampling procedure goes on, information becomes available. The importance sampling densityfunction can therefore be adapted during sampling, so-called adaptive importance sampling (Au

25

u1

u2

g1(u1,u2)<0

g2(u1,u2)<0

g3(u1,u2)<0

g4(u1,u2)<0

unsafe

unsafe

unsafe

unsafe

pf

Figure 2.6: Crude Monte Carlo

and Beck, 1999; Maes et al., 1993a, 1993b; Melchers, 1999). The more samples are taken, thebetter the sampling density function will approximate the optimal sampling function. A similartechnique is used in Directional Adaptive Response surface Sampling (DARS) and will bediscussed there, section 2.5.

Example Crude Monte CarloThe Monte Carlo method is performed on the complete system with its 4 limit state functions. Again theanalysis is terminated when a sufficient accuracy is reached. The samples are generated randomly,according to the standard normal distribution of the random variables. The number of limit state functionevaluations equals the number of samples (N). The results are given in Figure 2.6. The samples areshown on the left hand side, the evolution of the failure probability versus N on the right hand side.Besides the mean value, the 95% confidence interval is shown. The borders (full line) show theboundaries of the preset 95% confidence level: 95% CI(pf)=[0.0256;0.0735], according to Eq. 2.9. Theobtained mean value (bold line) indeed is enclosed by this interval representing the required accuracy: pf= 0.0644. The number of samples needed to reach this accuracy, amounts N = 298. The amount ofsamples is higher than the estimated number: 3/pf = 75.

26

u1

u2

g1(u1,u2)<0

g2(u1,u2)<0

g3(u1,u2)<0

g4(u1,u2)<0

unsafe

unsafe

unsafe

unsafe

pf

hV(v1,v2)

Figure 2.7: Monte Carlo - Variance increase

Variance increase Monte CarloTo restrict the number of samples, variance increase is used. The sampling function equals:

(2.23)( ) [ ]hV v = � ��

��

�

��Φ 0 0 13 0

0 13

2

2, ..

according to the following rule (Waarts, 2000):

(2.24)σ βh n= =−0 4 13. .

The number of samples is limited to: N = 37. The obtained failure probability equals: pf = 0.0503, seeFigure 2.7.

Importance Sampling Monte CarloImportance sampling has been performed too, Figure 2.8. Therefore, the samples were generated aroundone of the design points: . The effect is that the obtained failure probability is no longer[ ]u1 2 2* ,=a system value, but is only representative for a single branch. The number of limit state functionevaluations equals: N = 62. The obtained component failure probability: pf,1 = 0.016. This indeed is a goodestimate for the failure probability of limit state function g1, but is hardly a good value for the system failureprobability as can be seen by the preset boundaries (95%CI(pf)=[0.0256,0.0735]). This is because a prioriinformation was used in an inefficient way which guides the result to a single branch. In this example

27

u1

u2

g1(u1,u2)<0

g2(u1,u2)<0

g3(u1,u2)<0

g4(u1,u2)<0

unsafe

unsafe

unsafe

unsafe

pf

Figure 2.8: Importance Sampling Monte Carlo

however, the different branches have comparable importance. Using variance increase this is accountedfor. Using importance sampling in the design point of a single branch is neglecting the importance of theother branches. A weight function with 4 importance sampling functions in the 4 design points would givegood results, but lower efficiency.

2.7. Directional Integration and Directional (Importance) Sampling

2.7.1. Directional IntegrationThe idea of using polar coordinates in stead of the Cartesian coordinate system originates fromDeak (Deak, 1980). As the analysis is performed in the polar coordinate standard normal space(λ,θ), the unit vector θ defines the direction and a scalar λ the length of the vector in the standardnormal space. The basic reliability problem, Eq. 2.14, is translated into:

(2.25)( )[ ] ( )p P g f df = ≤ =λ θ θ θΘ Θ ΘΘ

0

where fΘ(θ) is the (constant) density of Θ on the unit sphere.

In case of Directional Integration (DI), for each direction θi the value of λi is determined forwhich the limit state function equals zero:

28

(2.26)( )g i iλθ = 0

In that, λi is a measure of the distance to the limit state in the direction defined by θi. Thedistance λi is found via an iteration procedure and requires several limit state function evaluations(LSFE). The average number of limit state evaluations is estimated to be 3 to 4 (Waarts, 2000),which is confirmed by the executed applications, see Chapter 6 and Annex A. The DirectionalIntegration method integrates over the directions θ. It follows from elementary probability theory(Benjamin and Cornell, 1970) that the radial distance λ is such that λ2 is chi-squared distributed(χn

2) with n degrees of freedom:

(2.27)( )( )( )p df n= −1 2 2χ λ θ θ

2.7.2. Directional SamplingIn case of Directional Sampling (DS), simulation using Eq. 2.25 proceeds by generatingrandomly N directional Monte Carlo simulations of the θ-vector. Each sample� , � ..., �,θ θ θ1 2 N

results in a sample value :�pi

(2.28)( )[ ] ( )� �p P gi i i n i= ≤ = −λθ χ λ0 1 2 2

An unbiased estimator of the failure probability pf can be written as the mean value of the samplevalues :�pi

(2.29)( )p E pN

pf f ii

N

≈ ==

� �1

1

The standard deviation can be estimated using (Ditlevsen et al., 1988):

(2.30)( ) ( )( )SN N

p E pp i fi

N

f

2 2

1

11

=−

−=

� �

When the number of samples N is sufficiently large, the estimator for pf is assumed to benormally distributed. This can be used to set up confidence intervals and to derive the requirednumber of samples necessary to achieve a preset accuracy. From numerical examples, it can beseen that the required number of samples N is function of the number of random variables n andof the resulting reliability index of the problem at hand. For a 95% confidence interval, β=4,n=100 it follows that approximately N=160n (Waarts 2000). As the average number of LSFEfor each sample equals 3, the total amount of LSFE is approximately equal to 480n. This isslightly higher than in the case of Monte Carlo Importance Sampling (300n). When comparedto the Monte Carlo technique it can be seen that it has similar advantages. It is a level III method,that accounts for the exact probability density function and system behavior. The limit statefunction must not be known explicitly, which makes it attractive for combination with finite

29

element analysis. Each sample requires a limited amount of limit state function evaluations(approximately 3 to 4 LSFE). These can be performed very fast in case the limit state functionis a simple functional form, which is the case for estimated response surfaces.

The difference between Directional Integration and Directional Sampling is illustrated in Figure2.9. The same example was used as in case of Monte Carlo simulation for ease of comparison.For a sufficiently high number N of simulations in case of Directional Sampling or a sufficientlysmall angle dθ in case of Directional Sampling, the results of course should be equal, theaccuracy as well.

2.7.3. Importance SamplingTo reduce the variance Spf

2, importance sampling can be introduced, similarly as in the case ofCrude Monte Carlo Simulation. The basic reliability problem, Eq. 2.2, is rewritten as:

(2.31)( )[ ] ( )( )p P g

fh

df = ≤ =λ φφφ

φφ

Φ Φ Θ

Φ

0

where hΦ(φ) is the importance sampling density function on the unit sphere. An unbiasedestimate of the failure probability is calculated from the sample estimates:

(2.32)

( )

( )[ ] ( )( ) ( )( ) ( )

( )

p E pN

p

p P gf

h

f

h

f f ii

N

i i ii

in i

i

i

≈ =

= ≤ = −

=

� � ,

� �

�

�

�

�

1

0 1

1

2 2λφφ

φχ λ

φ

φΘ

Φ

Θ

Φ

As before, the sampling efficiency will depend strongly on the chosen sampling density functionhΦ(φ) and a good choice requires prior knowledge on the phenomenon, which might not beavailable. In general, there are several methods of importance sampling that can be used incombination with directional sampling depending on the availability of prior knowledge:decrease of variance of unimportant variables, truncation of distribution functions of variableswith known importance to the limit state function, applying weight functions, skippingunimportant variables or shifting variables. Furthermore, depending on the choice of hΦ(φ), thecomputation of fΘ(θ)/hΦ(φ) might require considerable computational effort. Therefore,performing this operation for high n is discouraged (Bjerarger, 1988; Ditlevsen et al., 1988, 1990;Kijawatworawet, 1992; Moarefzadeh, 2000; Waarts, 2000). As a priori information is unlikelyto be available, Importance Sampling will not be used in this study.

30

u1

u2

g1(u1,u2)

g2(u1,u2)

g3(u1,u2)

g4(u1,u2)

unsafe

unsafe

unsafe

unsafe

pf

u1

u2

g1(u1,u2)

g2(u1,u2)

g3(u1,u2)

g4(u1,u2)

unsafe

unsafe

unsafe

unsafe

Directional Sampling Directional Integration

Figure 2.9: Directional Sampling and Directional Integration

ExampleDirectional SamplingThe number of samples (N) in case of directional sampling amounts 9, Figure 2.9. The number of LSFEis 33. Thus almost 4 limit state function evaluations are required for each direction to find the root λi. Theresults can be summarized as:

(2.33)p LSFEf = = = =0 0453 169 9 33. , . , , # N β

Directional IntegrationIn case of directional integration, following formulation is used:

(2.34)( )( )pf n i ii

≈ −=

(1 2

1

5

χ λ θ θ∆

31

where:

(2.35)∆θ π= 25

5 discrete segments are sufficient to obtain the preset accuracy, requiring 23 LSFE, see Figure 2.9. Theresults can be summarized as:

(2.36)p LSFEf = = =0 0443 170 23. . , #, β

The number of samples or LSFE is rather low. This is mainly because this example is very suitable to beprocessed in polar coordinates. All directions have a comparable contribution to the global failureprobability.

2.8 First order and second order reliability methods

2.8.1 FOSM - First Order Second Moment methodBecause of their simplicity, so-called 'second-moment methods' have become very popular sincethe work of Cornell (Cornell, 1969). Rather than to use approximate (and numerical) methodsto calculate the outcome of the general reliability problem, Eq. 2.2, in the following sections, theintegrand fX(x) is simplified. When using a second-moment method, only the first two moments -mean (µ) and standard deviation (σ) - are used to represent the random variables.

Assume that in Eq 2.1. the resistance R and the load S are second-moment random variables(having normal distributions). Then, the limit state function equals the safety margin (Z):

(2.37)g R S Z R S( , ) = = −

and the failure probability is:

(2.38)( )pfz

z

= −� � = −Φ Φµσ

β

in which µZ and σZ equal:

(2.39)µ µ µ σ σ σZ R S Z R Sand= − = +2 2

β is the safety or reliability index and Φ() the standard normal cumulative distribution, Figure2.10.

Evidently, Eq. 2.38 yields the exact failure probability pf when both R and S are normallydistributed. When this is not the case, pf defined in this way, is only a nominal value. Therefore,

32

β

µZ

σZ

pf

fZ(z)

Z=(x-µZ)/σ

Figure 2.10: The FOSM principle

the use of the reliability index is preferred, as it only refers to 'reliability' and 'safety'. β is a directmeasure of the safety of the structural element. Greater β means greater safety, or lower nominalfailure probability pfN (introduced for convenience and clarity).

This method is readily extendable to linear limit state functions g(X), with n (correlated) randomnormal variables:

(2.40)( )g a b X b X an nTX b X= + + + = +1 1 ...

The failure probability then reads:

(2.41)p af

T

T= − +�

���Φ b

b C bX

X

µ

where CX is the covariance matrix, accounting for the possible correlation between the randomvariables.

This also limits the range in which correct values for the failure probability are obtained. In casethe limit state function is no longer linear, which is likely to appear in practice, a linearisation

33

gL(X) can be performed around a preset point , using a first order Taylor expansion:~x

(2.42)( ) ( ) ( )g g gX

X xLii

n

i iX xx

= + −=

~ ~~

∂∂1

The resulting reliability index is therefore called a First Order Second Moment reliability index:βFOSM. Disadvantage of this βFOSM is that the result is not invariant of the chosen linearisationpoint nor of the mathematical way the problem is formulated, Figure 2.11.~x

Example - FOSMFOSM, FORM and SORM methods are level II methods, these are performed on a single branch only,namely:

(2.7)( ) ( ) ( )g u u u uu u

1 1 2 1 22 1 22 0 0 1

2, . .= + − −

+

For the FOSM method, the first order derivatives of the limit state function are required. These equal:

(2.43)( )

( )

∂∂∂∂

gu

u u

gu

u u

11 2

22 1

0 2 12

0 2 12

= − −

= − −

.

.

Depending on the linearization point the linearized limit state function results in (first order Taylor~uexpansion in the linearization point):

(2.44)[ ]~ ..

~

. .

.

u

u

11

2

1

2

1

2 875 20 375 2

0

2

01

2 875 2 0 375 201

0 375 2

= � � �

=

= −

�

���

���

�== −

���

�

= −

= −−�

�

�

=

∂∂∂∂

αα

β αgugu

T

(2.45)[ ]~.

.

.

..

~

...

.

u

u

21

2

1

2

2

2 20 663

0 274

114

0 2340 972

2 2 0 6630 2340 972

131

= � � �

= −

= −

�

���

���

�= −= −

���

�

= −

= − −−−

�

�

�

=

∂∂

∂∂

αα

β αgu

gu

T

34

ββββ1

u1ββββ2 ββββ3gL1

gL2

gL3

u2

u3

u1

u2

Figure 2.11: FOSM in different linearization points

(2.46)[ ]~ ..

.

.

.

.

~

. ...

.

u

u

31

2

1

2

3

15 20 913

0 47

0 95

0 440 90

15 2 0 9310 4410 898

175

= � � �

= −

= −

�

���

���

�= −= −

���

�

= −

= − −−−�

�

�

=

∂∂∂∂

αα

β αgugu

T

These points and the resulting reliability index are represented in Figure 2.11.

This problem was solved by Hasofer and Lind (Hasofer and Lind, 1974). First the normallydistributed random variables X as well as the limit state function are transformed into thestandard normal space (u-space): U, g(U)=0. The point in which the linearisation is performed,was chosen as the point for which a minimum distance β is found, further denoted as the designpoint or failure point u*. This point represents the point of greatest probability density or thepoint of maximum likelihood for the failure domain, Figure 2.12.

In the standard normal space the shortest distance and hence reliability index is given by:

35

u1

u2

ββββ=2

αααα1

αααα2

u*

Figure 2.12: FORM

(2.47)( )

( )

β = � � =

==

min min

:

u

subject to g

ii

nT2

1

12

12

0

u u

u

From the geometry of surfaces it follows that the direction cosines αi of the outward normal onthe hyper plane are, Figure 2.12:

(2.48)α

∂∂

∂∂

ii

ii

n

gu

gu

=

=

2

1 u*

With αi known, Figure 2.12 shows that the coordinates of the failure point or design point equal:

(2.49)ui i* = −α β

These direction cosines are of great interest. They represent the sensitivity of the standardizedlimit state function g(u)=0 at the failure point u* to changes in u (Bjerarger and Krenk, 1989;

36

Hochenblicher and Rackwitz, 1988) and are important in a double sense. In case the sensitivityαi of ui is low, there is little need to be very accurate about the determination of ui. Also it wouldsignal that, if necessary, ui might well be treated as a deterministic rather than a random variable,which could reduce the dimensionality of the space of random variables and thus computationaleffort. In case the sensitivity αi of ui is high, these are the variables that should be looked at withcare. The failure probability is sensitive for changes in their probability density function.

When the failure point u* is used as the expansion point in the first order Taylor expansion, Eq.2.42 finally becomes:

(2.50) ( )g ui ii

n

u = + ==

β α1

0

Most computational effort is put into the optimization algorithm to find the minimum distanceβ, Eq. 2.47. Since the point u* is not known a priori, the problem of finding the shortest distanceβ in u-space, subject to g(u)=0, is strictly a minimization problem. A general overview ofminimization procedures can be found elsewhere (Himmelblau, 1972; Judge et al., 1980).

These procedures can be divided in two subgroups: gradient methods and non-gradient methods.Gradient methods use less iterations than non-gradient methods and are used for this type ofproblem in general. However, these gradients have to be delivered at cost of analyticaldifferentiation (in case the limit state function is known analytically) or by means of numericaldifferentiation (at the cost of extra limit state function evaluations). As the limit state functionmight not be available analytically, analytical differentiation will not be used in this study. Somecomment can be found in literature (Waarts, 2000).

Although the global iteration scheme is the same (Melchers, 1999), several authors havesuggested procedures to improve the search for the design point. Initial gradient methodprocedures with slow convergence required a substantial amount of limit state functionevaluations (Hasofer and Lind, 1974; Rackwitz and Fieβler, 1978; Madsen et al., 1986).

The failure point u* is obtained, using a sequence of points u1, u2, ... according to the rule:

(2.51)u u dk k ks+ = +1 .

in which dk is the search direction and so the step length, that varies in between 0 and 1. In the

gradient projection methods, the kth direction vector dk, is given by:

(2.52)d u uk k k kT

k= −α α

in which αk is a unit row vector with first order derivatives, evaluated in the iteration point uk,according to Eq. 2.48. In case of numerical differentiation these require at least 2 LSFE.Because of its slow convergence, several LSFE are required.

Hasofer and Lind (HL-Hasofer and Lind, 1974;RF- Rackwitz and Fieβler, 1978) proposed a

37

method that was modified with a merit function by Der Kiureghian (Liu and Der Kiureghian,1986) to improve its convergence. The modified HL-RF method employs the direction vector:

(2.53)[ ]d u uk k k k kT

k= − −α α α

Instead of a constant step length so, the step length is adapted each iteration step (divided by 2 in

practice), such that for the merit function m(u) yields:

(2.54)( ) ( )

( ) ( )

m m

m g

k k

T

u u

u u u u

+ <

= − +

1

2 212

12

α α

Although the convergence is almost sure, the decreasing step length s results in slow convergenceand requires a higher number of iterations and thus LSFE.

Several authors have suggested to improve the search for the failure point. Well-knownalgorithms belonging to the group of efficient Sequential Quadratic Programming methods thatdo not need relaxation factors, are the so-called RFLS (Abdo and Rackwitz, 1990) and NLPQL(Schittowski, 1985) algorithms. These are implemented in commercial packages now-a-days(RCP, 1997). This results in a reliable, efficient and fast method to calculate the FOSMreliability index βFOSM.

However, several disadvantages remain:• the method only enables to calculate the exact failure probability for normally distributed

random variables;• the limit state function is linearized in the failure point u*, which leads to inaccurate

results in case of large curvatures;• only the component reliability is computed as the method is only capable of calculating

the contribution of a single point of maximum likelihood u*.

2.8.2 FORM - First Order Reliability MethodThe first difficulty of inaccurateness of the method in case of non normal distributions can beovercome by the so-called normal-tail transformation (Melchers, 1999). A transformation toequivalent standardized normal distributions is inserted in the iteration scheme whenever needed:

(2.6)( ) ( ) ( )( )Φ Φu F x u F xi X i i X ii i= ⇔ = − 1

As this extends the reliability method to more then 2 moments (µ,σ) solely, it is called FORM(First Order Reliability Method) is stead of FOSM. This step to a more accurate result does notrequire any extra limit state function evaluations and is therefore applied in all subsequentmethods when required.

38

Example - FORMFor the design point u* = [0.7071,0.7071] the first order derivatives, first order Taylor expansion andresulting reliability index become, see Eq. 2.42, Eq. 2.48 and Eq. 2.49:

(2.55)( ) ( )

u

u

u

u

u

*

.

*

*

*

= � � �

= −

= −

�

�

��

�

��

�= −

= −

�

���

���

= − +

= +=

22

222

2

22

22

2 0 22

1

2

1

2

1 2

1

2

∂∂∂∂

α

α

β α

gugu

g u u

u

L

i ii

(2.56)[ ]

β αFORMT= −

= −−

−

����

����

=

u*

.2 22

22

2

2 0

The direction cosines or sensitivities equal: α=(-0.7071,-0.7071). This means that both random variableshave equal importance to the failure probability. This is graphically illustrated in Figure 2.12. Dependingon the algorithm used, the computation requires 12 LSFE (RFLS) or 17 LSFE (NLPQL).

2.8.3 SORM - Second Order Reliability MethodTo account for possible curvatures (κ) of the limit state function in the neighborhood of thefailure point u*, Breitung (Breitung, 1984), proposed a second order approximation of the firstorder reliability index (βFORM):

(2.57)( ) ( )p pf f SORM FORMi

n

i≈ = − ∏ +� �=

− −

, Φ β βκ1

11

12

in which the curvatures κi are defined as the eigenvalues of Dn-1, the matrix with principalcurvatures of the paraboloid in the failure point u*: Hohenblicher and Rackwitz (Hohenblicherand Rackwitz, 1988) derive a similar, pre-asymptotic solution:

(2.58)( ) ( )( )p pf f SORM FORM

FORM

FORMi

i

n

≈ = − +� �

−

=

−

∏, ΦΦ

βϕ β

βκ1

12

1

1

39

In here, φ is the marginal normal probability distribution and Φ de cumulative normaldistribution function. Similarly as in case of the gradient methods, these second order derivativescan be determined analytically or numerically, depending on the availability of the functionalform of the limit state function.

A lot of effort is put in finding appropriate methods to avoid calculating the second orderderivatives, as it requires numerous LSFE. Discussion on the derivation of the second orderderivatives can be found in literature (Breitung, 1984, 1994: Ditlevsen and Madsen, 1996;Geyskens, 1993; Der Kiureghian et al., 1987). Der Kiureghian and Stefano (Der Kiureghian andStefano, 1991) proposed an iterative procedure that finds both the principle curvatures and thefailure point at the same time on an iterative basis, which is elaborated in more detail elsewhere(Schueremans, 1998).

In many cases the effort put in calculating these curvatures is much greater than performing aFORM analysis on the limit state function, followed by an Importance Sampling Monte Carlo,as the region of importance (the required a priori information) u* results from the FORManalysis (Hohenblicher and Rackwitz, 1988). This also holds for irregular limit states, althoughthese are assumed to be an exception in structural reliability. A first estimate of the reliabilityindex can be found using FORM/SORM analysis. An update of the analysis is obtained byISMC.

Example - SORMFor the SORM method, the curvatures are required. Therefore, the Hessian (A) with normalized secondorder derivatives is required in the design point u*:

(2.59)( )( )

( ) ( )

( ) ( )Auu

u u

u u=

∇∇

=����

����

=−

−�

��

��

2

2

21

2

1 22

2 1

2

22

0 2 0 20 2 0 2

gg

gu

gu u

gu u

gu

**

* *

* *. .. .

∂∂

∂∂ ∂

∂∂ ∂

∂∂

To obtain a convenient formulation for the parabolic approximation, the standard u-space is rotated untilu* is located on the nth (2nd) axis. Therefore an orthonormal transformation R is performed, with thedirection cosines α put in the last row:

(2.60)D RARnT

T

= = −− −� �

−−�

��

��

− −−

�

��

�� =

�

��

��

2 22 2

0 2 0 20 2 0 2

2 22 2

0 4 00 0

. .. .

.

The principal curvatures κi i = 1,...,n-1 can be determined as the eigenvalues of the matrix Dn, leaving outthe last row (Dn-1). In this case this simple solution leads to:

(2.61)κ 1 0 4= .

40

The principal axes formulation of the approximating paraboloid in the rotated u’-space reads:

(2.62)u u thus u un ii

n

i' ' , : ' . . '= + = +

=

−

β κ12

2 0 12

0 42

1

1

2 12

This was already visualized in Figure 2.5, where the analytical integration was performed. As the originallimit state function is a second order polynomial, a perfect match of the original limit state function isobtained.

Once the approximating paraboloid is constructed, the probability content of the bounded second-ordersurface can be determined, using Breitungs asymptotic approximation (Breitung, 1984), which is exact forβvQ:

(2.63)( ) ( )

( )( )

p pf f SORM FORM FORM ii

n≈ = − +

= − + =

−

=

−

−

∏,

* . .

Φ

Φ

β β κ1

2 1 2 0 4 0 01696

12

12

1

1

The corresponding second order reliability index (βSORM) equals:

(2.64)βSORM = 2127.

Using a similar, pre-asymptotic solution, derived by Hohenbichler and Rackwitz, the solution becomes(Hohenbichler and Rackwitz, 1988):

(2.65)

( ) ( )( )

( )( )

p pf f SORM FORMFORM

FORMi

i

n

SORM

≈ = − +� �

= − + =

=

−

=

−

−

∏,

. * . ..

ΦΦ

Φ

βϕ β

βκ

β

1

2 1 2 3732 0 4 0 016292 137

12

12

1

1

which is closer to the exact value, that yields:

(2.66)β = 2141.

In practice, most effort is put in calculating the second order derivatives. This proves to be a tedious jobin practice (Geyskens, 1993). Using numerical methods, the amount of LSFE (using RFHL-algorithm forFORM analysis) equals 26.

41

2.8.4 System Analysis (SA)FORM/SORM methods are unable to cope adequately with the case of piecewise limit statefunctions or more than one limit state function and therefore remain level II methods orcomponent level methods. To obtain a level III method or system level method for multiple limitstate functions using FORM/SORM methodology, the different component failure modes haveto be combined using a system analysis (Melchers, 1999), Figure 2.13.

Thoft-Christensen (Thoft-Christensen, 1985) proposed the Branch-and-Bound method. Thefailure probability of each failure mode has to be calculated, using FORM/SORM. Furthermore,the structure has to be re-analyzed after failure of one component. Basic problem in the analysisis to define all possible failure modes. The identification of the failure components is to bedecided by engineering judgement. For frame structures this might be acceptable, for continuousstructures as masonry walls, this is not a realistic alternative.

An overview of various methods is given by Kuschel (Kuschel et al., 1999). It is concluded thatthere is no generally applicable method to identify all failure mechanisms. The best strategydepends on the problem at hand, and in most cases is a combination of several methods, againat the expense of an increased number of LSFE.

When the different failure modes (F1,...Fm) are identified they can be modeled as a combinationof parallel and series systems.

In case of serial systems, a union of failure domains is calculated. Then, an upper- and lowerbound can be calculated by the Dilevsen bounds (Ditlevsen, 1979), Figure 2.13:

(2.67)( )[ ] ( )p P F F p p p P F Ff ii

m

j i j ii

m

f f f i i jj

i

i

m

, , ,max max ,= <= =

−

=− < < + −� �

���

��

�

�1 21

1

1

20� �

In case of parallel systems, an intersection of failure domains is calculated. Then, anapproximation is proposed by Hohenblicher (Hohenblicher and Rackwitz, 1983). Alternatively,the serial system bounds can be applied, using (Melchers, 1999):

(2.68)P F P Fi

m

ii

m

i= =

� � = − �

��

��

1 1

1� �

These bounds are believed to be close to each other in most cases. However, depending on thecorrelation between the different failure modes, this is not always the case (Melchers, 1999).

Another customary method (Ditlevsen et al., 1987; Lin and Der Kiureghian, 1987; Ditlevsen etal., 1990) to obtain the system failure probability is to perform directional sampling on thelinearised component limit state functions or to perform importance sampling Monte Carlo usinga multi-modal importance sampling density function.

Considering the additional complexity in identifying the different failure modes and thecomposition of the system behavior afterwards, a method that accounts for the system behavior

42

in se is preferred in case of complex or continuous structures.

Example - FORM/SORM and system analysisA system analysis is performed when the FORM component failure probabilities are available. For theserial system with its 4 branches, Figure 2.3, these equal:

(2.69)

pppp

f g FORM g FORM g SORM

f g FORM g FORM g SORM

f g FORM g FORM

f g FORM g FORM

, , , ,

, , , ,

, , ,

, , ,

. ; . ; .. ; . ; .. ; .. ; .

1 1 1

2 2 2

3 3

4 4

0 0228 2 00 2 140 0228 2 00 2 140 0062 2 500 0062 2 50

= = =

= = =

= =

= =

β ββ βββ

Because of the linearization in the design points and the small correlation between the different branches,the failure content in the intersection between the different branches is too small to have any influence onthe global failure probability. Therefore, the Ditlevsen system bounds are the same and equal the sum ofthe component failure probabilities, see also Figure 2.3, right hand side:

(2.70)p p p p pf FORM f g FORM f g FORM f g FORM f g FORM, , , , , , , , , .≈ + + + =1 2 3 4

0 058

Although this value differs from the correct failure probability (because of the curvatures), this value wouldalso be the result of Monte Carlo or Directional sampling on the linearised limit state functions.

The total amount of LSFE is the sum of the number of LSFE for the different branches. In case of FORMusing a RFLS-algorithm this amounts 36, in case of a NLPQL-algorithm, this yields 48.

Because of the positive curvatures, the intersection between the different branches will even decrease andwill have a negligible influence on the global failure probability. Again, the Ditlevsen bounds coincide:

(2.71)p p p p pf SORM f g SORM f g SORM f g FORM f g FORM, , , , , , , , , .≈ + + + =1 2 3 4

0 045

which of course is a good approximation of the real failure probability. The number of limit state functionevaluations equals 66.

To illustrate the Ditlevsen bounds to their full extent, the theoretical example is altered slightly. To increasethe correlation between the different failure modes, the sign of the curvatures is switched, their value isamplified and the component failure probability of the linear branches is decreased:

43

u1

u2unsafe

unsafe

unsafe

unsafe

safe

g1

g2

g3

g4

A

B g2L,B

g1L,A

g2L,A

g4L,B

ρ(g1LA, g2L,A)A=0.6ρ(g2L,B, g4L,B)B=0.83

Component reliability :

pf,g1=0.104 βg1=1.26pf,g2=0.104 βg2=1.26pf,g3=0.067 βg3=1.50pf,g4=0.067 βg4=1.50

System reliability:

pf=0.22 β=0.77

Figure 2.13: Altered limit state function and system analysis

(2.72)( )

( ) ( ) ( )

( ) ( ) ( )

( )( )

g u u

g u u u uu u

g u u u uu u

g u u u ug u u u u

1 2

1 1 2 1 22 1 2

2 1 2 1 22 1 2

3 1 2 1 2

4 1 2 2 1

2 0 0 252

2 0 0 252

15 215 2

, min

, . .

, . .

, ., .

=

= − − −+

= − − ++

= − += − +�

The altered limit state functions are shown in Figure 2.13. The component reliability indices and failureprobabilities for the different branches of the limit states as well as the global failure probability (pf) andcorresponding reliability index (β) are shown in Figure 2.13.

Remark that for this particular example, the second order reliability index is the same as the first orderreliability index because Breitungs’ approximation fails in case βFORM .κ=-1, which is the case for thebranches g1 and g2:

(2.73)β β β βg FORM g FORM g SORM g SORM1 2 1 22 00 2 00, , , ,. ; .= = = =

Furthermore, the global failure probability no longer equals the sum of the component failure probabilities:

(2.74)p p p p pf f g f g f g f g≠ + + + =, , , , .1 2 3 4

0 342

44

The failure probability is overestimated because of the correlation or overlap between the different failuremodes. For the calculation of the different intersections, the accuracy can be increased if the second orderbounds are based on a linearization performed in the point of intersection as illustrated for points A andB on Figure 2.13 (Van Dyck, 1995). For point A the correlation (ρ) between g1 and g2 equals: ρ(g1LA,g2LA)A=0.6. For point B the correlation between g2 and g4 even is ρ(g2LB,g4LB)B= 0.83. Then the system failureprobability according to the Ditlevsen bounds equals:

(2.75)[ ]pf = 0 186 0 251. ; .

2.9 Conclusions

The generalized reliability problem is outlined and an overview of traditional reliability methodsto solve this problem is given. Before going into detail on the different methods, some commenton the required accuracy is made. A new criterion is proposed, depending on the actualreliability level. The different methods are illustrated on an academic example, to demonstratetheir pro's and contra's.

Analytical and Numerical Integration (AN/NI) are level III reliability methods. Their majordrawback is the number of limit state function evaluations with respect to the number of randomvariables 9n. Therefore, these methods are mainly applied for validation other methods, usinga low number of variables.

Crude Monte Carlo (MC) is a level III reliability method. Major advantage is the simplicity ofthe method, which makes it very attractive to be used in combination with finite element analysisor response surfaces. Its major drawback, the high number of samples to reach the requiredaccuracy, can be overcome by using importance sampling (ISMC). For the case of varianceincrease (MC+VI), an experimentally determined sampling function is used.

Using polar coordinates leads to the analogous techniques, called Directional Integration (DI)and Directional Sampling (DS). The efficiency of the latter is comparable with MC+VI. Majoradvantage is the fact that the roots of the limit state function are calculated, which provideinteresting information about the structural behavior. On average, 3 LSFE are required to findthis root for each sample. Because the number of LSFE remains proportional with the numberof random variables, the efficiency is sufficient to be combined with a response surface.

First Order and Second Order Reliability Methods (FORM/SORM) are level II methods.Although these methods are very efficient, a System Analysis (SA) is required to obtain level IIIestimates of the failure probability. Because only a lower- and upper bound can be calculated,this limits their use in combination with other techniques, despite of their high efficiency.

The results based on the used academic example, are summarized in Table 2.2. Some overallevaluation is given, based on preset criteria (Waarts, 2000). These are derived from the examplestreated in Annex A.

45

Criterion

Che

ckpo

ints NI

MC

ISM

C

MC

+VI

DI

DS

FOR

M(+

SA)(

RFL

S)

FOR

M(+

SA) (

NLP

QL)

SOR

M(+

SA)

multiple critical points,unions and intersections

- ++ ++ - +/- ++ ++ -- --

number of n - -- ++ +/- +/- - - - -

probability level pf ++ ++ -- +/- +/- ++ ++ ++ ++

strong curvatures in LSF / ++ ++ ++ ++ ++ ++ -- +/-

discontinuous LSF / + + + + ++ + -- --

no roots in axis direction ++ ++ ++ ++ ++ + ++ ++ ++

implicit LSF ++ ++ ++ ++ ++ + + - -

number of LSFE 2n 9n 3/pf 300n 480n

number of LSFE fortheoretical example

4 81 298 62 37 33 23 36 48 66

level of reliabilitymethod

I III III II III III III II(III) II(III)

Legend: '++' behaves very well or has very little influence; '+' behaves good or has littleinfluence; '+/-' moderate; '-' behaves less or has influence; '--' behaves bad or has greatinfluence; '/' no information available

Table 2.2: Overview of reliability methods and their applicability [adopted from (Waarts, 2000)]

46

47

3 Reliability analysis using an adaptive responsesurface

3.1 IntroductionIn Chapter 2, focus was on traditional reliability methods. Some of their drawbacks have led tonew developments in which the original limit state function is replaced by an estimated responsesurface. These new developments are the main subject of this chapter.

The motivation for the use of a response surface and the response surface methodology itself aresubject of Section 3.1. The combination of the standard reliability methods and an adaptiveresponse surface is dealt with in Sections 3.2-3.4. From Chapter 2 it was concluded thatDirectional Sampling, Monte Carlo sampling with variance increase and FORM/SORM to alesser extent, are very well capable of calculating an accurate value for the reliability index.Therefore, these techniques are very promising to be combined with an adaptive responsesurface. Each of these combinations is worked out in a separate section.

Directional Adaptive Response surface Sampling (DARS) was first developed by Waarts(Waarts, 2000). The technique is described and improvements are added to increase theefficiency of this procedure.

Monte Carlo Adaptive Response surface Sampling with Variance Increase (MCARS+VI) ispresented in Section 3.3. Although the procedure is similar to DARS, some fundamentaldifferences exist. These are further investigated.

FORM/SORM does not to result in a system reliability value, which is a major disadvantage.Therefore, the technique is only mentioned for completeness. It will not be used in theapplications.

The applicability, limitations and accuracy of these methods with respect to the evaluation ofexisting masonry structures are discussed. For ease of comparison, the different methods areillustrated using the same mathematical example that was used in Chapter 2. To refine thedifferences between the DARS and MCARS+VI procedures, some academic examples areworked out. These are presented in Annex A.

3.2 Implicit limit states - Response surface methodology

Extremely idealized structural systems can be studied in an analytical way, both in thedeterministic and probabilistic case. A real structure however, has a high degree of mechanicalcomplexity. Therefore, even a deterministic analysis can only be conducted by numericalalgorithms, read: finite element analyses or the outcome is only available from experiments. Inthese cases the Response Surface (RS) method can offer an outcome. The main idea is that theresponse consisting of a complex function of input variables is approximated by a simplefunction of the input variables. When the Response Surface is capable of handling the complexstructural behavior, the reliability analysis can be performed on the Response Surface, in stead

48

of using the original problem:

(3.1)( ) ( )g gRSX X→

In that, the limit state function evaluations are simple mathematics on the Response Surface thatonly demand little computational effort. So, the type of reliability method used is of littleimportance since the time consuming LSFE (using FEM) are replaced by analytical expressions.Of course, several limit state function evaluations are required to build the Response Surface.An optimal scheme is required to obtain a sufficiently accurate response surface, demandingminimum limit state function evaluations.

The Response Surface method is gaining field in combination with finite element analysis(Bjørset et al., 1999; Bucher et al., 1988; Faravelli, 1989; Giannini et al., 1996a, 1996b; Roos etal., 1999; Schueremans et al., 1999; Venezianou et al., 1983). A general description of theResponse Surface methodology can be found elsewhere (Box and Draper, 1987; Breitung andFaravelli, 1996; Montgomery, 1997; Petersen, 1985; Schueremans, 1998; Yoa and When, 1996).

The standard reliability procedure using standard response surface methodology in combinationwith a finite element analysis reads:• select the most important random variables on the basis of engineering judgment;• make a choice of one of several families of functions, which appear to be suitable to

approximate the unknown system behavior;• make a design of experiments, which give optimal estimates of the parameters of the

chosen functional form;• construct a response surface through these response data;• validate the obtained response surface;• perform a reliability calculation on the response surface in stead of on the real response.

These steps are described in the following sections. Although this scheme replaces a number ofdirect limit state function evaluations, this is not an optimal scheme. Optimizing this scheme isperformed using Adaptive Response Surface techniques. This will be described in Sections 3.3-3.5.

3.2.1 Selection of random variablesAs already mentioned, for real structures, the number of variables can be very high. The majordisadvantage of the Response Surface method lies in the exponential growth (~2n) of the numberof experiments (and thus LSFE) with respect to the number (n) of random variables in the vectorX needed to estimate the response surface (in case of standard response surface methodology).Therefore, most important variables are chosen based on engineering judgment. These are placedin a subvector X1. The vector X of random variables therefore is subdivided into (Breitung andFaravelli, 1996; Giannini et al., 1996a, 1996b):

(3.2)[ ]X X X= 1 2,

This leads to:

49

Non controllable variables: X2

output: g(X)Controllable variables: X1system

Figure 3.1: General representation of a system - output (g) is function of controllable (X1) andnon controllable (X2) variables

(3.3)( ) ( ) ( )g gRS RSX X X= +1 1 0 2ε

in which ε0(X2) is an error term represented by a zero-mean normal distributed random variable,accounting for the exclusion of X2 in gRS1(X1). This intrinsic randomness is denoted the "pureerror".

This principle is derived from experimental practice. In case of real experiments on a system,some of the variables are controllable (X1), others are less or non controllable (X2). This isvisualized in Figure 3.1. The output g(X) of course is function of both.

In case of a displacement controlled compressive test on brick masonry walls (system), thedisplacement is the controllable variable, the force the output. Amongst the non-controllablevariables are: temperature, relative humidity, to what extent the upper- and lower edge of the testwall are rectified and the stiffness of the test bank used. In case of a reliability analysis, no experiments in the real sense are performed. Using MonteCarlo or Directional Sampling, sampling is performed. Therefore, splitting up the randomvariables X into two sub-sets is artificially and only done to limit the number of samples requiredto obtain accurate results. Using standard Response Surface methodology without splitting upthe random variables, can only be performed on systems with a limited number of randomvariables (n) because of the exponential growth of samples required to estimate an accurateresponse surface (~2n). Adaptive Response Surface sampling meets this disadvantage to a largeextent. Using an adaptive response surface scheme, the number of limit state functionevaluations is limited to approximately 15n and remains proportional to the number of randomvariables (Waarts, 2000). As this is judged to be acceptable, a large number of variables isprocessable. Splitting up the random variables will no longer be required.

For clarity, the general formulation will be presented in the following sections. In case ofAdaptive Response Surface sampling, section 2.6, the subdivision will no longer be used, thus:X=[X1], X2= [ ].

To perform fitting in the x-space is an arbitrary choice. Fitting in any other space, mostfrequently the u-space is used, may perform as well.

50

3.2.2 Response Surface - functional formThe response surface is generally constructed by fitting a polynomial of low order to the samplingpoints. Depending on the problem at hand, a linear or quadratic function is appropriate in mostcases. To account for the interaction between different variables, cross terms can be included.In the most general case the response surface looks like:

(3.4)( )g X X XRS i ii

n

ij i jj i

n

i

n

X1 0 1 11

2 1 11

= + += ≥=

θ θ θ, , , , ,

Because the functional form will not cover the real behavior perfectly, a lack of fit error-term isintroduced εL(X1). Accounting for the limited number of random variables that is includeddirectly, the real structural behavior is given by:

(3.5)( ) ( ) ( ) ( )g gRS LX X X X= + +1 1 0 2ε ε

Of course, both error terms can be merged into a residual error εR, see Section 3.2.4. Eq. 3.5 istransformed into matrix notation for ease of handling:

(3.6)( ) ( ) ( )g T TLX X X X X X= + + + +θ θ θ ε ε0 1 1 1 2 1 1 0 2

And further into:

(3.7)( ) ( ) ( )g LX A X X= + +θ ε ε1 0 2

in which A is denoted the observation matrix that contains the linear, quadratic and cross-terms.

Given ne experiments with outcome the leastx x x1 2, ,..., ne( ) ( ) ( )[ ]g x x x= g g g n1 2, ,...,

square estimates of the regression coefficients θ are obtained using (Petersen, 1985):

(3.8)( )�θ =−

A A A gT T1

and the estimated outcome, based on the estimated response surface yields:

(3.9)� �Y A= θ

It is evident that the choice or design of experiments is of major importance.

51

3.2.3 Design of ExperimentsTarget of the design of experiments is to perform a series of experiments able to estimate theregression coefficients θ in an optimal way. A first strategy could consists in changing a singlevariable into its high or low value in each experiment and keeping the other variables constant,so xi=(0,...,±1,...,0). In practice this means, all values are represented by their mean value, exceptone:

(3.10)( ) ( ) ( ) ( )( )x i j j nx x x x= ±µ µ σ µ1 1 1 1 1 1, , , ,,..., ,...,

This method has the important disadvantage that possible interaction between different variablesis lost.

The correct way is to change different variables at the same time using an optimal factorialdesign. One of the most frequently used schemes to change different variables at the same time,is the Central Composite Design (CCD), Figure 3.2. The Central Composite Design has threemajor parts. The first number of experiments (2n) are a full factorial 2n design. These arerequired to estimate a linear response surface, extended with cross-terms. To account for higherorder terms, star points (2n) are added. Introducing central points (n2), it is possible to accountfor the pure error (εe). The number of central points has to be chosen by the experimenter. Anumber n2 of 5 to 8 experiments is considered acceptable (Breitung and Faravelli, 1996).

The distance αrot for the star points is chosen by the rotatability requirement. A rotatable designis obtained when αrot yields (Montgomery, 1997):

(3.11)( )α rotn= 2

14

Other optimal experimental designs do exist, such as Box Behnken Design, the Face-CenteredDesign. For more information, the reader is referred to standard work on this matter(Montgomery, 1997).

Example - RS-methodThe Response Surface method is illustrated using a single branch solely, namely g1. Both variables areincluded in X1, thus will be accounted for directly. A second order polynomial, with cross terms is used toestimate the response. The outcome of the experiments, following a Central Composite Design, is usedto estimate the regression coefficients. To meet the requirement of rotatability, αrot equals:

(3.12)( )α rot = =2 221

4

Based on the response surface a FORM algorithm is used to find a first estimate of the reliability index andfailure point u*RS,1, Figure 3.2.

52

CCD :Center point : [0,0]

Axial points:

[ ][ ][ ][ ]

1010

0 10 1

,,

,,

−

−�

Star points:

[ ][ ][ ][ ]

2 2

2 2

2 2

2 2

,

,

,

,

−

−

− −�

gRS,1= 2.0-0.47u1u2

-0.43u12-0.43u2

2

βRS,1,FORM = 1.73u*RS,1 =[1.22,1.22]

u1

u2

g1(u1,u2)=0

gRS,1(u1,u2)=0

u*RS,1

βRS,1,FORM

Figure 3.2: Response Surface methodThe first estimate βRS,1,FORM = 1.73 is an acceptable estimate for the real first order reliability index: β1,FORM= 2.00.

3.2.4 Validation of the Response SurfaceThe validation of the Response Surface is based on an analysis of variance (ANOVA) (Wackerlyet al., 1996). This enables to estimate the variances of the error-terms εL and εe. This issummarized in Table 3.1.

Type of variance Sum ofSquares

degrees offreedom

Mean Squares andVariances

F-value

model lack of fitpure errortotal error

SSLSSESSR

ne-n2n2-1ne-1

MSL= σεL2 = SSL/(ne-n2)MSE = σεe2 = SSE/(n2-1)MSR = σεR2 = SSR/(ne-1)

F=MSL/MSE

; ; ; ;( )SSR y yii

ne

= −=

2

1

y yii

ne

==1

( )SSE y yi ni

n

= −=

2

2 2

1

yn

yn ii

n

2

21

2 1

==

SSR = SSE + SSL, Legend: SSL: Sum of Squares of lack of fit; SSE Sum of squares ofpure error; SSR: Sum of Squares of total error; MSL: Mean squares of lack of fit error;MSE: Mean squares of pure error; MSR: mean squares of total error

Table 3.1: Variance analysis - validation of response surface - summary

53

Based on hypothesis testing, the validity of the response surface is checked. The null-hypothesisand the alternative hypothesis are (Breitung and Faravelli, 1998):

H0: the model is correctH1: the model is incorrect

The test statistic that is used:

(3.13)F MSLMSE

=

is F-distributed (Van Dyck and Beirlant, 1995), with ν1=ne-n2 an ν2=n2-1 degrees of freedom.Disagreement with the null hypothesis is indicated by large value of F; hence the rejection regionfor a test with significance level α is given by:

(3.14)F F> α ν ν, ,1 2

3.2.5 Design updatingThe design of experiments is made around a chosen center point. Normally, the mean value ofthe random variables is chosen as first center point: xc= µ(x). As the response surface isavailable, a first estimate of the design point x1* can be obtained, using a FORM/SORMmethod. In case of a single failure mode, the contribution to the failure probability is the greatestin the neighborhood of the failure point (Melchers, 1999). A new response surface can beconstructed, using the first estimate of the failure point as new center point: xc,2= x1*. Thisprocess (xc,i+1= xi*) of course can be repeated until sufficient convergence is obtained, Figure 3.3.

Example - RS-method - Design updateThe design is updated in the estimated failure points u*RS,1. This is used as a new center point: uc,2 =u*RS,1, Figure 3.3.

54

CCD:centerpoint : *, ,u uc RS2 1=

gRS,2= 2.0-0.70u1-0.70u2-0.20u1u2+0.10u1

2

+0.10u22

= gu1

u2 gRS,2(u1,u2)=g

u*RS,2=u*

βRS,2,FORM

Figure 3.3: Response Surface Method - Design update

As a result, the original limit state function is found, as this is only a second order polynomial.

3.3 DARS - Directional Adaptive Response Surface Sampling

The previous chapter indicated that a fast reliability method should be searched for in directionalsampling, Monte Carlo Importance Sampling or FORM/SORM. To decrease the number ofdirect limit state function evaluations, these can be used in combination with the ResponseSurface methodology as described, although the calculation scheme should be adapted to preventthe exponential growth (~2n) as a function of the number of random variables n. Using a CCDindeed 2n+2n samples and thus LSFE are required to estimate the response surface. For a highnumber of random variables (n), this leads to an unacceptable number of LSFE. Therefore, anadaptive response surface technique in combination with Directional Sampling is described inthis section.

The DARS method (Directional Adaptive Response surface Sampling) is a combination of thedirect and indirect method (Waarts, 2000). It can be looked at as an algorithm minimising thenumber of direct limit state function evaluations.

55

3.3.1 DARS - Governing relationsThe DARS procedure is performed in different steps, Figures 3.4-3.6, illustrated for the standardnormal space, u-space.

• Step 1. In a first step the value of each random variable is increased individually untilthe root (λ) of the limit state is found. This can be seen as an Axis Directional Integration(ADI) procedure, section 2.4. First a linear estimate is calculated, based on the outcome(LSFE) in the origin of the u-space (standard normal space) (0,0,0,…,0) and the point(0,0,…,±β0,…,0). When known, the start value β0 is set equal to the expected reliabilityindex β. Experience shows that β0=3 performs good as well. Further approximations arebased on a quadratic fit through the outcome (LSFE) of the different iteration points.Mostly after 3 to 4 iterations (LSFE) the root is found, assumed there is a root in thespecified direction. Each time extra verification is done to assure convergence to thecorrect root in the preset direction, Figure 3.4.

• Step 2. An initial response surface (gRS,1) is fit through the data using a least squarealgorithm, Figure 3.5.

• Step 3. This step is an iterative procedure. The response surface is adapted (AdaptiveResponse Surface method (ARS)) and the failure probability or reliability index areupdated until the required accuracy is reached, see section 2.3. Therefore, coarsedirectional sampling is performed on the response surface. For each sample, a firstestimate of the distance λi,RS to the origin in the u-space is made based on the responsesurface. An arbitrary distance λadd is used to make distinction between important and lessimportant directions: λi,RS <> λmin + λadd, in which λmin is the minimum distance found sofar. When λadd is set equal to 3, a sufficient accuracy is reached within a relatively smallnumber of samples. The value λadd = 3 is an arbitrary value that is optimized for the caseof probabilistic design, where target βT values around 3.8 are aimed at. In case theobtained root (λi,RS) has a relatively high contribution - λi,RS < λmin + λadd - to the estimatedglobal failure probability (pf), the root of the real response is used in stead of the responsesurface: λi,LSFE. This requires several extra limit state function evaluations. The responsesurface is updated with these new data from the moment they are available. In case thecontribution is less important - λi,RS > λmin + λadd - the contribution based on the root of theresponse surface (λi,RS) is used, avoiding time consuming limit state function evaluations(LSFE), Figure 3.6.

It is proved that the expected value of all contributions ( ) is an unbiased estimate( )( )E pf� �pi

of the global failure probability pf (Melchers, 1999; Waarts, 2000):

(3.15)( ) ( )p E pN

p where pf f ii

N

i n i RS or LSFE≈ = ==

� � , : � ,1

1

2χ λ

The major advantage of the method lies in the number of direct limit state function evaluations.These remain proportional to the number of random variables (n). Waarts estimated that thenumber of direct limit state function evaluations is limited to approximately 15n for anacceptable accuracy: V(β) = 0.05. Again this value is for an estimated target reliability indexβT=3.8. (Waarts, 2000). This linear increase gains interest for high numbers of n. In addition,

56

Step 1: Evaluate the LSF inthe origing of the u-space [0,…,0].Search the roots λ ofthe limit state functionfor the principaldirections[0,0,…,±β0,0,…,0]

u1

u2

[ββββ0,0] λ=λmin

[0,ββββ0]

[-ββββ0,0]

[0,-ββββ0]N = 5 = 2n+1#LSFE = 21λmin = 3.5β=2.85

Figure 3.4: DARS - Step 1

there is no preference for a certain failure mode. All contributing failure modes are accountedfor, resulting in a safety value that includes the system behaviour, thus on level III.

This procedure runs in an integrated pilot version of the generic finite element code Diana 7.1(Diana, 1998; Waarts, 2000). For purpose of the present research, the DARS procedure wasimplemented in Matlab 5.3 (Mathworks, 1999). In combination with an automatic interface withthe Calipous program (Smars, 2000) this provides a powerful tool to calculate the reliability ofmasonry arches at level III, see Application 3. Similarly, in combination with the finite elementcode Calfem (Calfem, 1999) an integrated stochastic finite element method (SFEM) is developed,Application 4 and Annexe A.15.

Example - DARS The DARS method is illustrated for the full serial system with its 4 branches. In Step 1 the factor β0 is setequal to 2.0, the expected reliability index. The required number of LSFE amounts 20 for finding the 4roots, so 5 LSFE for each single root. This is mainly because the absolute tolerance is set relatively small:

(3.16)( )∆g g= − ≤u 0 0 04.

After this step of Axis Directional Integration (ADI), the minimum distance λmin = 3.05. Thus the minimumdistance to the limit state function g1 is not found yet. A first estimate of the reliability index and failureprobability can be calculated. The reliability index equals β=2.85, the corresponding failure probabilityyields: pf = 0.0022, Figure 3.4.

57

Step 2: Fit a response surfacethrough these data

u1

u2

gRS,1= 1.65-0.13u1

2

-0.13u22

gRS,1 = 0

λmin = 3.5λadd = 3.0

Figure 3.5: DARS - Step 2

A first estimate of the response surface is fit through the data, Figure 3.5.

In the iteration procedure, Step 3, coarse directional sampling is performed on the response surface, untilthe preset accuracy is reached. The total amount of samples yields: N = 14. These required 51 LSFE (onaverage 3.6 LSFE to find the root). Thus, the response surface delivers a good estimate for the roots ofthe real response. After each sample, the response surface is updated when new LSFE are available.The resulting response surface is shown in Figure 3.6. The obtained failure probability and reliability indexequal:

(3.17)p LSFEf = = = = =0 028 191 51 14 2 05. , . , # , .min N and β λ

Remark that the minimum distance still has to be found. The obtained failure probability lies between thepreset borders of the 95%CI(Pf)=[0.0235,0.075], Figure 3.7.

58

Step 3: Perform DirectionalSampling on theResponse Surface:

If λ i,RS < λmin+λaddCalculatepi(LSF)=χ2(λ i,LSF,n)Update the responsesurface with new data

ElseCalculatepi(RS)= χ2(λ i,RS,n)

u1

u2

gRS,2= 0.92+0.046u1 -0.023u2-0.074u1u2 -0.097u1

2-0.084u22

gRS,2 = 0

λmin = 2.05λadd = 3.0

Figure 3.6: DARS - Step 3

Of course, a more accurate result will be obtained when the number of samples is increased, Figure 3.7.The obtained response surface is getting closer to the real response which results in better estimates forthe roots of the LSF in the sample directions. The results can be summarized as follows, Figure 3.7:

(3.18)p LSFEg

f

RS

= = = = =

=

0 050 164 184 520 2 00

3

. . , # , .min

,

, , N 0.35 - 0.0052u + 0.0042u - 0.017u u - 0.050u - 0.055u1 2 1 2 1

222

β λ

59

u1

u2

gRS,3 = 0

pf N = 14

λadd=3 λmin=2

Figure 3.7: DARS - increased accuracy

The number of samples, required to reach the preset accuracy is marked on the right hand side of Figure3.7 and equals N=14.

3.3.2 DARS - ValidationThe response surface outcome does not coincide with the real system behavior, as mentionedbefore. Performing a reliability analysis on the response surface therefore introduces a modeluncertainty, that can be adjusted for by introducing a lack of fit error term, εL. This is only thecase when the reliability analysis is performed on the response surface. When the DARSprocedure is performed, the reliability analysis is not only based on the response surface. In theimportant regions, where λ < λmin+ λadd, the real response is used, as the LSFE is accounted for.In those regions, no error is made. In the regions that are less important, λ > λmin+λadd, thereliability analysis indeed is performed on the response surface. In that case, no extra LSFE isperformed, which in fact is the gain of the method. In these points however, the outcome is notknown. Therefore, in these regions, the error can not be evaluated as this would require extraLSFE. Although the outcome of the response surface might not be accurate, its contribution tothe failure probability is small any way. So, the error made will always be small for a sufficientlylarge value of λadd. Of course, the higher value of λadd, the higher the number of direct LSFE.

However, a reliability analysis on the response surface might be interesting: to look for thevariables that contribute most in reaching the failure point x* or to estimate the effect of adaptingthe variable’s mean or standard deviation, in case more information becomes available, seeApplication 3. Therefore, it is interesting to check how good the response surface estimates the

60

real system behaviour. An hypothesis test based on paired observations, can be used to checkthe difference in outcome using both methods on the same samples (response surface outcomeversus real system behaviour based on the real LSFE) (Van Dyck and Beirlant, 1997; Wackerlyet al., 1996). When looked at from another point of view, an analysis of variance is performed.This will be outlined here to illustrate the analogous with the former variance analysis, section2.5.4. The hypothesis test of course results in completely similar results. The extra computationrequired is limited to the calculation of the response surface outcome for those data-points inwhich a LSFE was already performed (which restricts the analysis approximately to the regionin which λmin < λmin + λadd). This amount of computational effort is limited, seen the simplefunctional form of the Response Surface.

The outcome is considered to be the outcome of two different treatments: Y1 is the outcome ofthe real response (LSFE), Y2 is the outcome of the response surface. The total variation (SSR)of the response measurements about their mean for the two samples is quantified by:

(3.19)( )SSR y yijj

n

i

LSFE

= −==

2

11

2

where the average of the 2nLSFE yields:

(3.20)yn

yLSFE

ijj

n

i

LSFE

===

12 11

2

This quantity can be partitioned into two parts:

(3.21)( ) ( )SSR n y y y yLSFE ii

SSL

ij ij

n

i

SSE

LSFE

= − + −= ==

2

1

2 2

11

2

� ��� ��� � ��� ���

where the mean of each treatment yields:

(3.22)yn

y for iiLSFE

ijj

nLSFE

= ==

1 1 21

, : ,

Similar as in Table 3.2, a variance analysis can be performed, Table 3.3.

Then MSL/MSE has an F-distribution with ν1=1 numerator degree of freedom and ν2=2nLSFE -2denominator degrees of freedom (Wackerly et al., 1996). Again, disagreement with the nullhypothesis is indicated by large value of F; hence the rejection region for a test with significancelevel α is given by:

(3.23)F F> α ν ν, ,1 2

61

A significance level of 95% will be preset.

Type ofvariance

Variance(Sum ofSquares)

degrees offreedom

Mean Squares and Variances F-value

treatment errorpure errortotal error

SSLSSESSR

12nLSFE-22nLSFE-1

MSL= σεL2 = SSL/(1)MSE = σεe2 = SSE/(2nLSFE-2)MSR = σεR2 = SSR/(2nLSFE-1)

F=MSL/MSE

Table 3.2: Variance analysis - DARS validation - summary DARS-Validation of RSBased on the N=14 samples, the obtained response surface is validated using an analysis of variance.The results are summarized in Table 3.3.

Type of variance Variance(Sum ofSquares)

degrees offreedom

Mean Squares and Variances F-value

treatment errorpure errortotal error

SSL= 0.06SSE= 32.2SSR= 32.2

12nLSFE-2= 992nLSFE-1= 100

MSL = SSL/(1) = 0.06MSE = SSE/(2nLSFE-2) = 0.325MSR = SSR/(2nLSFE-1) = 0.322

F=MSL/MSE = 0.19

Table 3.3: Variance analysis - DARS validation - example summary

The rejection region for a test with 95% significance is given by:

(3.24)F F> =α ν ν, , .1 2

4 96

The F-value does not belong to the rejection region, the null hypothesis has to be maintained. Theobtained response surface is a good estimate of the real response.

3.3.3 DARS - Remarks• ADI-samples: The reliability indexes based on the ADI results sometimes differ a lot

from the final results (Waarts, 2000). This is mainly because the ADI is limited to a fewdirections. This difference is often seen when only a small set of random variables arereally important in the failure modes. This is also seen in the applications, seeApplication 3 (arches). This problem can be overcome by not taking these contributionspi into account for the estimate of pf. If all random variables contribute equally, thesesamples can be included in the reliability analysis.

• Antithetic directional sampling: For some limit state functions, only the positive θ ornegative -θ direction is of importance to the limit state function. Therefore theperformance of the directional sampling procedure can be improved accounting for theseantithetic directions. For this purpose, for each simulated direction θ, a positive and

62

negative distance (λ+ and λ-) are computed. The probability contribution pi, Eq. 3.15, isreplaced by the average:

(3.25)( ) ( )

pp p

ii i=

++ −λ λ2

According to Deak (Deak, 1980), the resulting variance is smaller than when usingone direction and twice the number of simulations. According to Waarts (Waarts,2000) the number of samples reduces to approximately 70%. The reduction in limitstate function evaluations however will be less, as it will cost an extra amount ofLSFE to search for both roots, for both opposite directions. As the limitstate functions are not known beforehand, neither their negative correlation, antitheticdirection is not used in this study.

• Truncation of variables - Conditional Sampling: When a priori information isavailable with respect to the importance of the random variables, this can be inserted too.This reduces the number of samples required significantly. For strength values forexample, it is not likely that higher strength values are important to the limit statefunction. This can be taken into account and sampling is only performed on the lowerhalf of the distribution function. Therefore a condition number (n_cond) is added to eachrandom variable. Possible values are [-1,0,+1]:• -1: sampling in the lower half of the distribution function,• 0: no a priori information,• +1: sampling in the upper half of the distribution function.The reduction of number of samples required equals: 2n_cond£0, which is certainly notnegligible. This possibility is implemented in the reliability procedure. It can be usedwhenever expedient.

3.3.4. DARS (λλλλadd = var)The number of LSFE that will be performed in the directional sampling procedures clearlydepends on the arbitrarily chosen λadd. Waarts (Waarts, 2000) made a comparison for λadd equalto 1.1, 3.0 and 5.0. Of course, the number of LSFE evaluations will be least when λadd = 1.1. Infact, the additional distance should be higher when a relatively bad response surface is fit to thedata. The lack of fit error is a measure for the goodness of fit and can be used to adjust theadditional λadd. In case the F-test rejects the response surface used, a higher value for λadd wouldbe appropriate. Naturally, if λadd vQ, a series of LSFE is performed for each sample. Then themethod relapse into crude Directional Sampling, leading to an optimal result, but with limitedgain in efficiency. The outcome of the response surface then is used as first iteration point in theroot finding algorithm. The number of LSFE in case of DS using an adaptive response surfaceis about 92n. Using λadd = 3, this number is limited to approximately 15n (Waarts, 2000).Therefore, λadd = 3, is proposed by Waarts. Waarts already suggests in his final conclusions thatthe additional distance λadd should be linked to the error between the roots of the real LSFE andthe roots of the response surface.

Furthermore, Waarts does not take all LSFE into account to estimate the response surface, onlythe roots. Intermediate results are omitted. This option is based on the idea that the response

63

g u u u u u uRS = + + + + +θ θ θ θ θ θ1 2 1 3 2 4 1 2 5 12

6 62

linear

interaction

pure quadratic

full quadratic

surface should not be an optimal representation of reality in all regions, but should only enableto make a proper distinction between the safe and unsafe region. In this thesis, a similarapproach is used for DARS (for MCARS+VI all LSFE are taken into account to estimate theresponse surface as there is no root-finding in MC sampling on purpose, see Section 3.7).

To increase the efficiency of the procedure, the value of λadd is related to the goodness of fit ofthe estimated response surface at hand. For each direction (i) for which the root in the u-spaceis calculated using the real limit state function, the root is also calculated in the u-space using theestimated response surface. Remark that this does not require any extra limit state functionevaluation:

(3.26)εroot i LSFE i RS iroot root, , ,= −

Each direction for which a new series of LSFE are calculated and a root is found, the regressionparameters are updated (+ an optimal choice of the regression model is made) and the errors εrootare recalculated for all roots. The computational effort required, remains relatively limited as thenumber of LSFE is assumed to remain limited. Furthermore, the gain is larger when a singleLSFE requires substantial amount of computational effort (such as in the case of non linear finiteelement analysis). The error εroot is taken as measure for the distinction between the importantand unimportant region. To be sure to encapsulate the error made by the functional form used,the 99% confidence interval is taken as measure for λadd:

(3.27)( ) ( )( ){ }λ µ ε σ εadd root n rootabs troot

= ± ×−

max . ,0 01 1

In here, t0.01,nroot-1 is the 0.01 quantile of the Student distribution, with nroot-1 (the number of rootsminus one) degrees of freedom (Van Dyck and Beirlant, 1999). Until the first 2n+1 roots arefound, the additional distance is kept λadd = 3.00, to prohibit a zero error by coincidence as onlyaxial points, thus without interaction between the variables, are looked at. Each iteration in step3 of the analysis, the response surface is updated, using a least squares analysis. To improve theaccuracy, the procedure itself chooses the functional form to be used from a family of secondorder polynomials. Distinction can be made between following forms (represented for n=2standard normal distributed random variables):

(3.28)

64

For these different models, the difference between the real root according to a series of LSEE andthe estimated root, based on the RS, is calculated: εroot,model_ i. The different models are comparedand the model leading to minimum error is used.

To illustrate the gain in efficiency that is obtained by relating the additional distance to theaccuracy of the estimated response surface, the examples that are used by Waarts to assure theefficiency of the DARS procedure, are recalculated. These results are added in Annex A. Theseexamples cover a wide variety of possible limit state functions. They were developed to comparethe different reliability methods with respect to some preset criteria (Waarts, 2000):• multiple critical points,• noisy boundaries,• unions and intersections,• space dimension,• probability level,• strong curvatures in the limit state function,• no roots in the axis direction. ______________________________________________________________________________Example - DARS λλλλadd = var - system with 4 branchesThe use of a variable additional distance is illustrated for the mathematical example with 4 branches. Inthis particular example the different directions have a rather equal importance, thus there will be no gainin limit state function evaluations. But, the evolution of the value of the additional distance λadd versus thenumber of limit state functions, is remarkable. Step 3 in the DARS procedure using λadd=var, is outlinedin Figure 3.8. The evolution of the variable distance λadd is shown in Figure 3.9. The final additionaldistance, λadd,fin, at the end of the iteration procedure (Step 3), is significantly lower than the initial value,that is taken 3.0.

65

Step 3 λλλλadd=var : Perform DirectionalSampling on theResponse Surface:

If λ i,RS < λmin+λaddCalculatepi(LSF)=χ2(λ i,LSF,n)• Update the RS• Update λadd

ElseCalculatepi(RS)= χ2(λ i,RS,n)

u1

u2

gRS,2= 0.92+0.046u1 -0.023u2-0.074u1u2 -0.097u1

2-0.084u22

gRS,2 = 0

λmin = 2.05λadd = 2.34

Figure 3.8: DARS (λadd = var) - Step 3 in the reliability procedure

Finally, the additional distance λadd,fin reduces to about 0.90. Initially, the value is taken equal to 3.0, thedistance proposed by Waarts. Initially, the value decreases, then again, increases up to approximately5.5. This is because a large error is observed between the roots of the RS and the real LSF. This is trueas can be seen from Figure 3.8. There is a small region in which the distinction between the important andunimportant area is made incorrect. The roots of the RS are considered in the unimportant region,although the real roots are in the important region. When the number of samples increase, this error isfound gradually. The response surface slowly converges to a shape that represents the real LSF moreaccurately, Figure 3.9. From that point, the error between RS and LSFE is limited. This can be observedin the evolution of λadd. From on a certain point, (nLSFE=64), the error in the distinction between unimportantand important region is no longer present. Remark that in the initial phase, this error has only smallconsequences to the resulting reliability index. This is because all directions have more or less equalcontributions to the failure probability. Moreover, this error is also made when the additional distance istaken constant and equal to λadd = 3.0.

66

u1

u2

gRS,3 = 0

λλ λλ add

N = 14,#LSFE = 51

λadd=0.89 λmin=2

nLSFE

Figure 3.9: DARS (λadd = var), final RS and λadd (left); λadd versus nLSFE (right)

__________________________________________________________________________________

3.4 MC or FORM and an Adaptive Response Surface (ARS)

3.4.1 Monte Carlo and Adaptive Response Surface sampling (MCARS)The former sections showed that Monte Carlo Importance Sampling is approximately equallyefficient compared with crude Directional Sampling. Therefore, in combination with an adaptiveresponse surface, this looks to be a very promising technique. Without major changes, thetechnique described in the previous section, can be combined with Monte Carlo sampling aswell: only samples close to a relevant limit state are computed using the real limit state functionand all the others are computed by means of the response surface.

The procedure is very similar to DARS. In fact, only the third step is altered (Waarts, 2000):• Step 1: The first step in the procedure is equal to the one in the DARS procedure. The

roots of the limit state function are searched on all axes (ADI), Figure 3.11.• Step 2: A first response surface is built on the basis of ADI results, Figure 3.11.• Step 3: The third step again is an iterative procedure. Variance Increase Monte Carlo

sampling is performed. The outcome based on the Response Surface is calculated. If thesample or its outcome is judged to be outside the important region, then the outcomebased on the Response Surface is used. If, on the other hand, the sample or its outcome

67

is in the important region, a new LSFE is performed. Again, the RS is updated as soonas new data are available.

Although the procedure looks very similar to DARS, some differences are apparent.

Because the use of a response surface already directs the LSFE samples to a certain (important)region, this is denoted as a kind of importance sampling. The samples in the non-importantregion are computed by means of the response surface and only cost minor computational effort.Since the response surface already directs the LSFE sampling, the use of extra importancesampling may be redundant and left out (Waarts, 2000).

This is only partly through. Crude Monte Carlo sampling lacks efficiency as the majority of thesamples - for normal practice with low failure probability - will be generated in the safe area.Although these do not require extra LSFE because most of them are outside the important region,the efficiency can be improved easily using Variance Increase Monte Carlo (MCARS+VI). Forthe sampling function fh(h), following empirical relation is proposed (Eq. 2.22,Waarts, 2000):

(3.29)σ βh n= −0 4.

The initial value for β is taken from the ADI procedure, step 1 and 2 in the analysis.

To distinguish the important from the non-important region, two different options can be taken.These are visualized in Figure 3.10. First, an approach similar to DARS is proposed, Figure3.10. In doing so, some difficulty arises which is inherent to the Monte Carlo methodology.Using Monte Carlo, random samples are generated and the outcome is calculated which doesonly require a single LSFE. No root finding algorithm has to be implemented. This means thatthe roots of the limit state function at hand are not searched for on purpose. Following procedureis used, Figure 3.10:• Not only the outcome based on the Response Surface is calculated, but also the direction

and distance (λ i) to the origin in the standard normal space (u-space), which does notrequire extra LSFE. In analogy to the DARS procedure, a region around the roots of theresponse surface is considered as important region. If λ i is outside the important region(λRS ± λadd/2), then the outcome based on the Response Surface is used. If, on the otherhand, λ i is within the important region (λRS ± λadd/2), then a LSFE is performed. Again,the RS is updated as soon as new data are available. Because no roots are calculated, itis not straightforward to link the calculated outcome with the additional distance (λadd)to be used. Therefore, a constant value is proposed, equal to the value used in DARS:λadd = 3.0.

68

u

gLSFRS

λλλλRS

λλλλadd

∆∆∆∆g,add

∆∆∆∆g,add

λλλλ i

gRS,i

gLSF,i

εεεεg,i

Figure 3.10: Distinction of the (un)important region in u-space and outcome space

Remark that, in opposition to the DARS procedure, all LSFE are taken into account and nolonger only the roots as these are no longer calculated on purpose.

A second approach is to rely on the outcome of the LSF itself to distinguish the important fromthe non-important area, Figure 3.10. Following procedure applies:• If the outcome of the RS is within a preset measure (∆g,add), the sample is considered in

the important region. A new value of the outcome is calculated, based on the real LSF.The response surface is updated with the new sample. If the sample of the RS is outsidethe preset measure, the sample is considered outside the important region. Again, theoutcome of the response surface is relied on in this case.

This second approach has the advantage that the difference (εg,i) between the outcome based onthe RS (gRS,i) and real LSF (gLSF,i) can be used to adapt the criterion itself that distinguishes theimportant from the non-important region, Figure 3.10. Following criterion is proposed:

(3.30)ε g i LSF i RS ig g, , ,= −

(3.31)( ) ( )( ){ }∆ g add g n gabs tLSFE, . ,max= ± ×−µ ε σ ε0 01 1

In here, t0.01,nLSFE-1 is the 0.01 quantile of the Student distribution, with nLSFE-1 (the number ofLSFE minus one) degrees of freedom.

69

No research has been focused on MCARS so far. In the example the method is worked out asillustration. As the method is similar to the DARS procedure, its efficiency is believed to bealike (Waarts,2000). This is not completely true.

Advantage of the DARS procedure is that it finds the roots of the limit state function, whereMCARS+VI only returns a positive or negative value of the limit state function. The roots ofthe limit state function contain interesting information with respect to the failure modes of thestructure. On the other hand, MCARS+VI does not require any complex root finding algorithm.Therefore, the procedure remains very simple and straightforward.

This directly influences the efficiency of the procedure used. For complex structures, a low orderpolynomial might no longer be efficient to capture the overall behavior of the structure, whereasit might be sufficient to represent the roots. The latter procedure is used in DARS. In this case,DARS will be more efficient compared to MCARS.

In other cases, where the overall behavior can be estimated accurately, using a low orderpolynomial, MCARS will be more efficient. This is because MCARS uses all LSFE to estimatethe regression coefficients of the response surface. Omitting the intermediate results in DARS,is omitting information. More LSFE will be required to fit a second order polynomial.

The gain in using a variable additional distance, as well in the standard normal space (u-space)as in the outcome space (x-space), depends on the structural system at hand. The gain will belarge, provided that the used response surface is capable of making an accurate distinctionbetween the safe and unsafe region. The gain will be less or negligible when this is not the case.

The efficiency in limit state function evaluations of the different methods is illustrated.Therefore the examples used by Waarts (Waarts, 2000) are re-analyzed using DARS andMCARS+VI. Both criteria for the distinction between important and non-important regions areprovided. These results are added in Annex A. In Application 3, Chapter 6, it is used tocalculate the safety of masonry arches. Example - Monte Carlo Adaptive Response Surface SamplingThe Monte Carlo Adaptive Response Surface Variance Increase sampling is performed on the serialsystem with 4 branches. The first and second step are identical to the DARS procedure, Figure 3.11.Initially, the important region is defined by λadd. Similar to DARS, λadd is taken: λadd = 3.0. This results inan accurate estimate with a relative low amount of direct LSFE. Depending on the method used, λadd iskept constant or ∆g,add is taken as measure to distinguish the important from the unimportant region. Bothare illustrated in Figure 3.12

70

u1

u2

gRS,1 = 0

λadd =3.00 λRS

Step 1: Search the roots λ ofthe limit state functionfor the principaldirections.(identical to DARS)

N = 2n+1 = 5#LSFE = 21

Step 2: Fit a response surfacethrough these data.(identical to DARS)

λadd = 3.00

Figure 3.11: MCARS+VI - Step 1 and Step 2 - λadd = 3.0

In the third step, Variance Increase Monte Carlo sampling is performed. When the sample is outside theimportant region defined by λadd, the outcome of the RS is used. Within the region of importance, a newLSFE is performed and the response surface is updated. This is presented in Figure 3.12.

71

u1

u2

gRS,2 = 0 (λadd=3.0)

pf

λadd

Legend :‘+’ = based on LSFE, ‘o’: based on RS N

gRS,2 = 0 (∆g,add=var)

λadd=3.0

∆g,add=var

Figure 3.12: MCARS+VI - Step 3 - λadd = 3.0 or ∆g,add = var

The results are summarized in Table 3.4. A good estimate is obtained within a relatively small number ofLSFE. Similarly, the results of the method using a variable distance in the outcome space (∆g,add = var)are listed. From Table 3.4 it can be seen that in this particular case, using a variable distance in theoutcome space does not lead to an efficient procedure. As mentioned before, all LSFE are taken intoaccount for the Response Surface estimate. The overall behavior of the system with its 4 branches cannot be described accurately using a second order polynomial. The final additional distance (at N=297samples), taken on the outcome values amounts: ∆g,add,fin = 1.51. This is a large value, comparable withthe maximum value of the limit state function (in the origin) gLSF(0,0)=2.0. Thus the major part ofevaluations will be in the important region, leading to a low efficiency.

MCARS+VI (λadd = 3.0) MCARS+VI (∆g,add = var)

pf β NnLSFEλadd//∆g,addRS-modelF/Fα

0.0331.8415021+58=793.00full-quadratic0.08/3.92

0.0451.7029721+171=1921.51pure quadratic0.09/3.82

Table 3.4: MCARS+VI - comparison between λadd = 3.00 and ∆g,add= var

72

The final response surfaces that are obtained, Figure 3.12, with λadd = 3.00 and ∆g,add= var, read:

(3.32)( )( )

g 1.03 0.02u 0.01u 0.04u u 0.11u 0.11uRS,2 1 2 1 2 12

22λ add

RS g addg u u u u

= = − − − − −

= = − − − −

3 00

147 0 008 0 0005 0 18 0 182 1 2 12

12

.

var . . . . ., ,∆

3.4.2 FORM and an Adaptive Response Surface (FORMARS)Structural reliability can generally only be performed with FORM when a system analysis isperformed to combine the different failure modes, see section 2.8.4. This means that severalcomponent reliability analyses have to be performed. The FORM analysis can be performed onthe response surface easily. Problem using this approach is that the response surface might notbe accurate in the region of interest, namely around the design-point or failure point x*. Thismight have a large influence on the calculated reliability indices.

First suggestions to link FORM calculations on the response surface with extra LSFE on the realproblem were made by Rackwitz and Fieβler (Rackwitz and Fieβler, 1978) and Karamchandani(Karamchandani, 1987). Waarts proposes an alternative procedure (Waarts, 2000):

• Step 1. A starting point x1 is defined (this usually coincides with the origin in the u-space)

• Step 2. The gradients for all limit state functions (g) are determined numerically. Thisrequires n+1 LSFE. The gradients are computed numerically. The second point ischosen µ+β0σ.

• Step3. For each individual limit state function (k), a design-point is estimated using thestandard FORM procedure. The g-values of all limit state functions in all estimates ofthe design-points are evaluated. This brings the total number of LSFE to 1+n+k.

• Step 4. A linear, if possible a quadratic, including cross terms response surface isconstructed on the basis of the (1+n+k) results.

• Step 5. An iteration procedure is initiated, each iteration consisting of three sub-steps:(a) a standard FORM calculation based on the current (adapted) response surface,leading to a new set of k design-point estimates;(b) a LSFE in each of the new estimated design-points is performed for all k limitstate functions;(c) a new adapted response surface (ARS) is built, based on the extended database(with k new values).

• Step 6. After N iterations the number of LSFE equals: 1+n+Nk. The iteration processis terminated when the value of the limit state function in the design-point for all limitstates is sufficiently close to zero (when both response surface and LSFE are close tozero).

After the FORM procedure has been completed, a combination procedure has to be applied tocompute the system reliability. One of the major disadvantages of this method is that all failuremodes need to be known beforehand. If this is not the case, the procedure has to be repeateduntil all failure paths have been investigated, which can be performed using a Branch-and-Bound

73

method (Thoft-Christensen and Murotsu, 1985), useful in the case of frame structures, but lessfor continuum problems. Therefore, it will not be used further in this study.

3.5 Comparison with Level I reliability methods

Level I reliability methods only check whether or not the reliability is sufficient instead ofcomputing the real probability of failure (EC1, 1994; ISO 2394, 1998). Therefore, combinationsof variables are taken in such a way that the failure surface is likely to be outside the boundarydefined by the reliability index βT, Figure 3.13. In case of one resistance and one dominant load(R-S-problem), Eurocode 1 (EC1, 1994) imposes the limit state function to be checked in thepoint (-0.8βT, 0.7βT) for resistance R and load S respectively. In here, βT is the target safety value.For practical use, the checkpoints are translated into design values using the partial safety factorsformat according to (EC1, 1994):

(3.33)RS

d R T R

d S T S

= −= +

µ β σµ β σ

0 80 7

..

The failure probability is supposed to be sufficiently low if in neither point the value of the limitstate function is negative.

When the influence of the variables on the limit state function is not known a priori, which islikely to be the case for complex structures, both possibilities have to be checked. For the R-S-problem this results in the combinations: (0.8βT, 0.7βT), (-0.8βT, 0.7βT), (0.8βT, -0.7βT) and (-0.8βT, -0.7βT). In general, the total number of checkpoints equals:

(3.34)N n= 2

In practice however, only a few points are checked, based on engineering judgment and structuralunderstanding (Vrouwenvelder, 1999; Waarts 2000).

Example - Level I reliability for the system with 4 branchesThe Level I reliability for the system with 4 branches can easily be calculated. It is similar to calculatingthe star points for establishing the Response Surface, section 3.2.3. Assume (u1,u2) to represent the R and S respectively and the preset target failure probability equals: pfT= 0.01. The corresponding target reliability equals: βT = 2.3. The checkpoint becomes, according to Eq. 3.33:

(3.35)( ) ( )-0.8 , 0.7 = -1.84,1.61T Tβ β

This is visualized in figure 3.13.

74

u1=R

u2=S

[-0.8β,0.7β] [0.8β,0.7β]

[-0.8β,-0.7β] [0.8β,-0.7β]

βT=2.3

Outcome ofcheckpoints :

g(-0.8β, 0.7β) = 0.28g( 0.8β, 0.7β) = -0.3g( 0.8β,-0.7β) = 0.28g(-0.8β,-0.7β) = -0.3

Figure 3.13: Level I - reliability control in checkpoints

It can be seen that this point is in the safe region. It would wrongly be concluded that the structure issufficiently safe. Indeed, the global failure probability and corresponding reliability index equal: pf = 0.045and β = 1.71. This is because not all checkpoints were controlled. When the other 3 checkpoints arecalculated, it is clear that for two of them the limit state function is violated. Thus, it is decided that thestructure is unsafe. The actual safety of the structure however remains unclear. Level I methods do notanswer that question.

3.6 Conclusions

This chapter treats recent developments in reliability methods, where traditional methods,described in Chapter 2, are combined with an Adaptive Response Surface (ARS). First, theresponse surface methodology is outlined. Although this method is very promising, its majordrawback again is the high number of LSFE required to fit an accurate response surface (~2n).

To minimize the number of LSFE, the response surface methodology is combined withtraditional reliability methods. Waarts was the first to combine Directional Sampling with anAdaptive Response Surface (DARS). This technique is improved by changing the arbitrarilyadditional distance λadd=3.0 into a variable distance, based on the root error in the standardnormal space. Similarly, the type of response surface used, is based on minimum error. An F-test is used for validation of the response surface.

75

Analogous to DARS, Monte Carlo with Variance Increase is combined with an AdaptiveResponse Surface (MCARS+VI). Both options, an additional distance λadd=3.0 and a variabledistance in the outcome space ∆g,add are treated in detail.

The efficiency of DARS and MCARS+VI are comparable and depend on the capability of theused response surface in capturing the real system behavior. For more complex applications, theefficiency of MCARS is believed to be less than DARS. This is caused by the fact that in caseof MCARS, the response surface is constructed using all LSFE. For complex structures, a loworder polynomial might no longer be efficient to capture the overall behavior of the structure,whereas it might be sufficient to represent the roots. The latter procedure is used in DARS.

In other cases, where the overall behavior can be estimated accurately, using a low orderpolynomial, MCARS will be more efficient. This is because MCARS uses all LSFE to estimatethe regression coefficients of the response surface. Than, omitting the intermediate results inDARS, is omitting information.

Advantage of the DARS procedure is that it finds the roots of the limit state function, whereMCARS+VI only returns a positive or negative value of the limit state function. The roots ofthe limit state function contain interesting information with respect to the failure modes of thestructure. On the other hand, MCARS+VI does not require any complex root finding algorithm.Therefore, the procedure remains very simple and straightforward.

Finally, a comparison with level I methods, using checkpoints, is made.

Table 3.5 summarizes the results on the academic example for the different procedures outlinedin this chapter. Again, an overall evaluation is given, based on preset criteria. These are derivedfrom the examples treated in Annex A.

As the limit state function is available analytically, most methods are convenient in finding theglobal failure probability, also the traditional methods, Chapter 2. When this is not the case, andmoreover, when it requires substantial amount of computational effort, a combination ofDirectional Sampling or Monte Carlo with an Adaptive Response Surface are preferable.

In the applications, different methods will be applied depending on the problem at hand. Theirapplicability as summarized in Table 2.2 and Table 3.5 will serve as a guideline.

76

Criterion

Che

ckpo

ints

DA

RS

( λad

d=3.

0)

DA

RS

(λad

d= v

ar)

MC

AR

S+V

I (λ a

dd=

3.0)

MC

AR

S+V

I (∆ g

,add

=va

r)

FOR

MA

RS

multiple critical points,unions and intersections

- ++ ++ + + --

number of n - + + ++ ++ -

probability level pf ++ ++ ++ ++ ++ ++

strong curvatures in LSF / ++ ++ ++ ++ ++

discontinuous LSF / ++ ++ ++ ++ --

no roots in axis direction ++ + + ++ ++ ++

implicit LSF ++ ++ ++ ++ ++ -

number of LSFE 2n 15n / / / /

number of LSFE fortheoretical example

4 51 51 79 192 /

level of reliabilitymethod

I III III III III II(III)

Legend: '++' behaves very well or has very little influence; '+' behaves goodor has little influence; '+/-' moderate; '-' behaves less or has influence; '--'behaves bad or has great influence; '/' no information available

Table 3.5: Overview of combined reliability methods and their applicability

77

4 Structural modeling of masonry

4.1 Introduction

When an assessment - deterministic as well as probabilistic, thus on level I, II or III - isperformed, the limit state function is needed. EC6 requires that for each relevant limit stateverification, a design model shall be set up. In several cases the border between the safe andunsafe region can be expressed analytically, in other cases a numerical model or finite elementmodel will be necessary. These models should be an appropriate description of the structure,the constitutive materials from which it is made and the relevant environment or boundaryconditions. The study can be limited to that part that is relevant for the limit state at hand. Allappropriate actions and how they are imposed, should be accounted for.

EC 6 offers analytical models for unreinforced masonry subjected to vertical, shear and lateralloading. Similar models can be found in other national building codes for masonry (SIA V177/2,1995; CSA S304.1, 1994; NBN B24-301, 1980). The relevant part of these models is describedbriefly, to illustrate that some of these models have a limited area of applicability. In practice,structural behavior is observed to be more complex. A great deal of research has been conductedby various authors to refine these models, as to offer greater similarity with reality. When thedesign model at hand offers a good similarity with reality, it could be used in a probabilisticevaluation procedure when the (partial) safety factors are stripped of (Melchers, 1999).

The reliability procedures that are developed in the previous chapter, will be used in theapplications, Chapter 6. These examples are chosen to be representative for unreinforcedmasonry, mainly for the case of historical buildings. Historical masonry is designed to act incompression. Therefore, Section 4.2 is assigned to masonry in compression. First eccentricloading is described in detail. From that, arching follows automatically, Section 4.3.

Mainly two other types of structural elements are described in modern codes: walls acting againstlateral loading (wind) and shear walls. These are typical for modern masonry structures, usingrelatively slender elements. A lot of research is focused on these two types of load-cases in thelast decennium. In fact, they have led to the need for more accurate numerical modeling usingfinite element methods, which is described in Section 4.5. Besides a global description, attentionis paid to the material models, available in commercial finite element packages.

The main purpose of this chapter is twofold. First, it is meant as theoretical background for thematerial models that are used in the practical applications, Chapter 6. The prime aim is notcompleteness in description. Only these models are described in detail, that will be used in theapplications, Chapter 6. However, some reference is made to other possible methods, to placethe models used in their correct perspective. Secondly, this Chapter shows which materialproperties need to be determined experimentally, which is further worked out in Chapter 5.

78

Figure 4.1: Masonry arches and supporting columns as an illustration ofeccentric compressive loading [taken from (Pieper, 1983)]

4.2. Masonry subjected to a vertical loading

Historically, masonry constructions are designed to act in compression. A typical example of acombination of arching and columns is presented in Figure 4.1.

79

FVe=180 mm

dc

ft

FVe=50 mm

σ1σ2

e=150 mm FV

σ1

σ2

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Eccentricity e [mm]

e=d/6 e=d/2σ=fc

elastic areaNTM

elastic area

emax,pl

emax,el

σ=ood = 600 mm, w=600 mm, FV = 140 kNft = 0,28 N/mm2, fc = 4,26 N/mm2

non-cracked cracked-NTM failure

plastic area

elastic area -cracked σ>ft

e=172mmcracked -

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

FVe=272 mm

fc

dc dpl

dpl

FVe=270 mm

fc

σ1=0dc

FVe=262 mm

fc

σ1=0dc

FVe=262 mm

ft

dc

σ2

FVe=d/6 σ2σ1=0

FVe=150 mm

σ1=0

σ2

dc

FVe=180 mm

σ1=0

σ2

dc

(a) (b) (c)(d)

(e)(f)

(g)(h) (i) (j)

(a)

Com

pres

sive

Stre

ss s

[N/m

m²]

σ=ft

Figure 4.2: Masonry subjected to an eccentric load - stress distribution

4.2.1 Analytical models When masonry is subjected to a vertical load, it will seldom be applied centric. The eccentricity

80

will influence the stresses in the masonry to a certain extent. This is visualized in Figure 4.2(Schueremans and Van Gemert, 2000; Smars, 2000), for a rectangular cross section.

The compressive stress as a function of the eccentricity is shown. The first graph does notaccount on the (limited) tensile strength of the masonry in the stress distribution. The secondgraph does take the tensile strength into account. The material properties used in Figure 4.2 arethose obtained from the experimental research, Chapter 5. These models are further elaboratedin Application 2 and 3, Chapter 6, introducing random variables, not only for the loads andmaterial parameters, but also for the geometry. The stress distribution for the elastic, cracked,plastic and intermediate stadia are described in detail. These are also shown in Figure 4.2 (a-j).

In the ideal case, when the vertical load (FV) is applied centric (e=0), the stresses are distributeduniformly over the cross section of the column. When the eccentricity remains limited (e<d/6),the stresses are distributed linearly over the cross-section, Figure 4.2(a), according:

(4.1)σ1 2 1 6, =

ױ� �

Fw d

ed

V

As long as the vertical load applies within the mid third area (e<d/6 for a rectangular crosssection), each point of the cross-section will be in a compressive regime. On the edge, when theeccentricity equals the mid third section (e=d/6), the compressive stress at one of the borders isreduced to zero, Figure 4.2(b):

(4.2)σ

σ

1

2

02

=

=×F

w dV

A further increase in eccentricity will induce tensile stresses, Figure 4.2(c):

(4.3)

σ

σ

1

2

1 6

1 6

=×

−� � ≤

=×

+���

��

Fw d

ed

f

Fw d

ed

Vt

V

Further increase of the eccentricity results in an increase of compressive stress which is morethan proportional. These stresses act on a decreasing non-cracked cross-sectional area. In casethe limited tensile strength (ft) of masonry is not accounted for - so-called NTM or Non TensionMaterial - the stress distribution will alter, Figure 4.2(d, f). The stress distribution will be suchthat an equilibrium is installed with the external force, only inducing compressive stresses:

81

FV

e

dc

ft

a/2ab

d/2-e

Ft Fc

σ

o

Figure 4.3: Stress distribution elastic-cracked-ft

(4.4)

σ

σ

1

2

02

32

=

=−� �

F

w d e

V

If the masonry is assumed not to sustain any tensile strength, cracks will occur. The crack depth(dc) can be calculated following:

(4.5)d e dc = −3

2

As the tensile strength of masonry is limited, the induction of tensile stresses in the masonry, Eq.4.3, will be limited. Cracks will occur after all. A stress redistribution will take place until theinternal stresses again are in equilibrium with the external load, Figure 4.2(e). This stressredistribution can be calculated based on the force and momentum equilibrium, Figure 4.3.

(4.6)

F F F F

M F d e F a F a b

i V c t

o i V c t

= ⇔ − + =

= ⇔ − −� � + − +���

�� =

0 0

02 2

32

0,

in which:

82

(4.7)

( )

( )

F w a Ff

F f w b f Ff

ab f

c cc V

c t

t tt V

c t

c

t

= =−

= =−

=

σ σσ

σ

σ

34

34

2

2 2

2

2 2

With some algebraic manipulation, Equation 4.6 and 4.7 can be transformed into:

(4.8)( )( ) ( )

a Fw f

e d f a ff

f

c V

c t

c t ct

c t

t

c

=−

−� � − + −−

+�

��

�� =

43

2 232

0

2 2

2 2 22

2 2

σσ

σ σσ σ

An iterative method is required to calculate the solution: compressive stress σc and the distancea. When these are found, the crack depth again can be determined, using:

(4.9)( )d d a bc = − +32

This process continues until the compressive strength of the masonry is reached, Figure 4.2(g,h).

In case of historical masonry, the weak but plastic mortar may deform plastically or the stonesmay be crushed until the complete cross-section is loaded up to its maximum load, Figure 4.2(i,j). This phase is not accounted for in the elder Belgian National Building Code, NBN B24-301(1980). The European Design Code EC6 (EC6, 1995) on the contrary accounts for an elastic-plastic behavior of masonry in compression. In practice, it has been recorded in some stronglydeformed arches (Smars, 2000). In case no mortar joint is in between the bricks/stones, thestrength of the bricks is normative. For the case of natural stones with high compressivestrength, this may result into higher allowable eccentricities. Again the stress distribution canbe calculated based on the force and momentum equilibrium. The tensile strength is no longeraccounted for, its contribution will be very limited as can be seen from Figure 4.2(g).

83

Fc,2eFc,1e

FV

e

fc

d/2-e

a b

o

Figure 4.4: Stress distribution elastic-plastic-cracked-ft

The force and momentum equilibrium yield, Figure 4.4:

(4.10)( )F F F F

M F d e F a b F bi V c c

o i V c c

= ⇔ − + =

= ⇔ −� � − +���

�� − �

��

�� =

0 0

02 3 2

0

1 2

1 2

, ,

, , ,

where:

(4.11)F f w a

F f wb

c c

c c

,

,

1

2

2=

=

This results in:

(4.12)

b Ff w

a

F d e a f w abf w b f w

V

c

Vc c c

= −

−� � − + +�

��

�� =

2

2 6 2 20

2 2

This quadratic problem can be solved analytically, resulting in:

84

(4.13)

a Ff w

d e Ff w

b Ff w

Ff w

d e Ff w

V

c

V

c

V

c

V

c

V

c

= × −� � −�

��

��

�

�

��

���

�

�

��

= − × −���

�� −

�

��

�

�

�

�

��

�

��

�

�

��

�

242

12

62

12

2 0 5

2 0 5

.

.

The elastic area has a depth a, the plastic area has a depth b. The crack depth (dc) thus equals:

(4.14)( )d d a bc = − +

Finally, the maximum eccentricity is reached when only a full plastic area remains. The plasticarea (dpl) and crack depth (dc) equal:

(4.15)

σc c

pl

c

f

d d e

d e

=

= −� �

=

22

2

Finally, the maximum force that can be transferred on a full plastic area, equals:

(4.16)σ2

22

= =−� �

f F

w d ec

V

The safety of an eccentrically loaded masonry column is further developed in Application 2,Chapter 6. The influence of the different models, described above, will be discussed.

4.2.2. Vertical loading - Design model according to EC6According to EC6 (1995), the resistance of unreinforced masonry walls to vertical loading shallbe based on the geometry of the wall, the effect of applied eccentricities and the materialproperties of the masonry. Following assumptions are allowed:• plane sections remain plain,• the tensile strength of masonry perpendicular to the bed joints is zero,• the stress/strain relationship is elastic-perfectly plastic.

Also, following items should be accounted for:• long term effect of loading,• second order effect,• eccentricities calculated from a knowledge of the layout of the walls, the interaction of

the floors and the stiffening walls,

85

• eccentricities resulting from construction deviations and differences in the materialproperties of individual components.

At the ultimate limit state (ULS), the design vertical load on a masonry wall, FV,Sd, shall be lessthan or equal to the design vertical load resistance of the wall, FV,Rd, such that:

(4.17)F FV Sd V Rd, ,≤

An acceptable design method for the verification of the ultimate limit state is given in EC6. Thiscan be seen as an elastic-plastic verification assuming NTM material for masonry.

Indeed, following expression is given for design resistance (EC6, 1995):

(4.18)F F

d fV Sd V Rd

k

m, ,≤ =

−� �2

1 2

γ

This results in:

(4.19)σγc d

V Sd k

m

Fd e

f,

,=−� �

≤2

2

Which is identical to the limit in case of an elastic-perfectly plastic analysis of a masonry wall,see Eq. 4.16.

4.3 Masonry arches

4.3.1. Thrust lines Columns and walls act in compression by developing vertical compressive stresses according tothe vertical forces that are applied as described above. Arches, vaults and cupolas withstandvertical loading by developing compressive stresses along their shape (Beckmann, 1994), Figure4.5. In that, the models as described above can be adopted when the position of the thrust lineis known. Indeed, in practice most arches are built with stones or bricks, with or without mortarjoints. Therefore, the material behavior is similar, although stones might have a significantlyhigher compressive strength compared to bricks.

86

Block A

r

Block A

Thrust line 1 : Fc,1 e=0.425d

Thrust line 2 : Fc,2

Figure 4.5: Possible thrust lines in a masonry arch and resulting stresses

The safe theorem for an arch reads (Der Kooherian, 1952; Heyman, 1966): “If a thrust line canbe found which is in equilibrium with the external loads and which lies wholly within themasonry, then the structure is safe”. This theorem already suggests that more than one solutioncan be found, Figure 4.5. Indeed, any thrust line that satisfies the safe theorem requirements, issufficient to ensure stability. It even does not have to be the actual line of thrust.

Two extreme lines of thrust can be found, in which three hinge points are formed. Because ofits degree of static undefinedness, three hinges are required to transform the structure into a staticstructure. Hinges are formed when the thrust line becomes a tangent line with the intrados orextrados of the arch (Lemaire and Van Balen, 1988).

From the former section it can be concluded that the stresses in the masonry are mainly dictatedby the ratio e/d and the compressive load Fc. For masonry arches, this ratio is often taken asmeasure to provide sufficient safety. In practice, the thrust line is limited to 5% of the border ofthe masonry arch to guarantee a sufficient safety: e = 0.450d. This means that 90% of the actualsection is taken as useful (Heyman, 1980). This maximum eccentricity initially was developedfor gothic arches, where the strength of the stones never was a problem. In case of heavily loadedarches or arches built from weaker stone material or bricks, the maximum allowable eccentricitymight be decreased to 7.5% (e=0.425d) to prohibit the crushing of masonry (Beckmann, 1994),Figure 4.5, dashed line. For a given load and known thickness of arch, the maximum eccentricitycan be calculated when the compressive strength is known, based on Figure 4.2 and the materialmodels behind this figure. Not accounting on the plastic behavior of the masonry and for thestrength given in Figure 4.2, the maximum eccentricity equals: e = 262 mm, or e=0.437d.

87

Fc 0.9d

0.1d

qlat

0.1d

0.9d

Fc

fk/γm

Plan view Detail A

wall

Arching betweensupports

2/31/3

L

d

Figure 4.6: Arching in case of lateral loading between supportscapable of resisting an arch thrust

Accounting on the plastic behavior, the maximum allowable eccentricity becomes e=272mm,or e = 0.453d. From the first value it can be concluded that the criterium for weak stones shouldbe applied. From the second value it is seen that the plastic behavior extends the allowableeccentricity to the same level as used for stone material.

The safety of masonry arches is further developed in Application 3, Chapter 6.

4.3.2 Arching - Design model according to EC6 Masonry arching is dealt with in EC6 in the framework of walls subjected to lateral loading, seealso Section 4.4. When a masonry wall is built solidly between supports capable of resisting anarch thrust, the wall may be designed assuming that an horizontal or vertical arch develops withinthe thickness of the wall. The rise of the arch is taken 0.9d, Figure 4.6.

For the verification of wall, accounting for arching between supports, the calculation should bebased on a 3-pin arch and the bearing at the supports and at the central hinge should be assumedat 0.1 times the thickness of the wall (d). Remark that this is a similar approach as presented inSection 4.2 and Section 4.3.1 (Beckmann, 1994; Heyman, 1980).

Than, the limit state according to EC6, equals:

(4.20)W W W f dLSd F F Rd

k

m

= ≤ = � �γγ

2

From Eq. 4.4, the maximum compressive force (Fc) can be calculated for the case of non tensionmaterial (NTM), Figure 4.6, detail A:

(4.21)F d f dfc

k

M

k

M

= × × =3 012

1510

. .γ γ

In case of three hinges, the maximum horizontal compressive force Fc in the mid point can be

88

Shear wall

Wind

Wall subjected to lateral loadingFigure 4.7: Lateral loading and shear walls [adopted from (Hendry, 1998)]

calculated, using (Van Gemert, 1995a):

(4.22)F q Ldc

lat=2

8

From Eq. 4.21 and Eq. 4.22 the design lateral strength is obtained:

(4.23)W W W q f dL

f dL

f dLSd F F Rd lat d

k

m

k

m

k

m

= ≤ = = � � ≤ × ���

�� = �

��

��γ

γ γ γ,. .

2 2 215 810

12

When the analytical solution is compared with Eq. 4.20 proposed by EC 6, it can be seen that anextra safety factor of 1.2 is implicitly present on the design lateral strength. Another implicitsafety factor is taken from the assumed eccentricity of the thrust line. As the central hinge isassumed at 0.1 times the thickness, the corresponding eccentricity equals e=0.400d, which isrelatively small, compared to the values obtained in Section 4.3.1 (e=0.437d, e=0.453d). Boththese conservative assumptions, will lead to a conservative reliability value, when used in aprobabilistic assessment, because the distinction between safe and unsafe is biased.

4.4. Lateral loading (wind) and shear walls

For unreinforced masonry, mainly two other types of loading are described in modern designcodes: walls withstanding lateral loads (wind) and shear walls, Figure 4.7. EC6 provideselementary design models for these two types of structural elements.

4.4.1. Unreinforced walls subjected to lateral loads - design model according to EC6The design of masonry walls subjected predominantly to lateral wind loads (γFWk) may be based

89

on approximate methods (EC6, 1995). The first is a wall supported along whichever edges, thesecond is based on wall arching between supports. The latter is treated in Section 4.3.2 as it isthe general approach in which arching is dealt with in EC6.

For the verification of walls, supported along the edges, the design moment MSd, shall be lessthan or equal than the design bending resistance (MRd):

(4.24)M MRd Sd≤

Masonry walls are not isotropic and there is an orthogonal strength ratio depending on the brickunit, mortar and pattern used (rµ), Figure 4.8.

When the failure is perpendicular to the bed joints, (fxk2 applies, Figure 4.8 (b)), the designmoment (MSd) should be taken:

per unit height of the wall (4.25)M r W LSd F k= α γ 2

When the failure is parallel to the bed joints (fxk1 applies, Figure 4.8 (a)), the design moment (MSd)should be taken:

per unit length of the wall (4.26)M r r W LSd F k= µ α γ 2

where:• rα is a bending moment coefficient depending on the orthogonal ratio rµ, the degree of

fixity at the edges of the panels and the height to length ratio of the panels and which isobtained from a suitable theory. These values are given in the National Applicationdocument (NAD) (NBN ENV 1996-1-1 NAD, 1998; Pfefferman et al., 1999),

• rµ is the orthogonal strength ratio of the characteristic flexural strengths: rµ= fxk1/fxk2,• γF the partial safety factor for loads,• Wk the characteristic wind load per unit area,• L the length of the panel between the supports.

It is clear that the main complexity is in establishing correct values for the bending momentcoefficient rα, which is ought to be filled in by a suitable theory. In the Netherlands, a researchproject was set up to obtain a better understanding of the real behavior and to derive materialproperties that can be used in numerical models. Parallel, finite element models were furtherdeveloped as it was believed that available numerical models lack similarity with reality (CUR171, 1994; Van der Pluijm, 1999).

90

(a) (b)

fxk1

fxk2

Figure 4.8: Flexural strength in orthogonal directions along and parallel to the bed joints [testsetup according to NBN B24-301 (1980)]

The design moment of lateral resistance is given by:

(4.27)( )

Mf I

vRd

xk

M

=γ

where:• fxk is the characteristic flexural strength, appropriate to the plane of bending based on

experiments and• I/v is the section modulus of the panel.

The above mentioned method mainly applies for infill panels. For load bearing structuralmasonry also acting in bending (due to wind load), the flexural strength is not a problem.Because of the compressive stresses from upper parts such as the roof, the wind load will seldominduce tensile stresses in the masonry:

(4.28)σ cvF

AMI v

= + ≥ 0

A similar thought can be made for masonry basement walls where the stress state is influencedby the horizontal ground stresses. Because of the upper structure, the compressive stressdominates the structures in such a way that tensile stresses are exceptional (Schueremans andVan Gemert, 2000).

91

VSd

VSd

NSd

lc

l

α

h

Figure 4.9: shear walls and lc of the compressed part of thewall [in accordance with SIA V177/2 (1995)]

4.4.2. Unreinforced masonry shear walls - design model according to EC6Resistance to horizontal actions is generally provided by a system formed by the floors and shearwalls. For the verification of shear walls (Ganz and Thurlimann, 1988), the design value of theapplied shear load, VRd, shall be less than or equal to the design shear resistance, VRd, such that:

(4.29)V VSd Rd≤

The design shear resistance is given by:

(4.30)V f dlRd

vk c

m

=γ

where lc is the length of the compressed part of the wall, ignoring any part of the wall that is intension, Figure 4.9.

This part is determined according to (SIA V177/2, 1995):

(4.31)( ) ( )l l h where VNc

Sd

Sd

= − × =2 tan , tanα α

and is calculated assuming a triangular stress distribution. In general, a linear elastic analysismay be used (EC6, 1995). To account for the advantage of possible redistribution of forces dueto limited cracking of the wall at the ultimate limit state, the shear on a single wall may bereduced by 15% provided that the shear on the parallel walls is increased correspondingly toassure equilibrium under the design loads. Walls that satisfy the ultimate limit state may bedeemed to satisfy the serviceability limit state.

92

The characteristic masonry shear strength is determined experimentally and varies with thevertical prestress (σv), according to the Mohr-Coulomb relationship:

(4.32)f cv v= +σ ϕtan( )

Several relationships are given in the standard, depending on the type of brick and mortar used(EC6, 1995).

From experimental research, Section 5.6, it is observed that the structural behavior is morecomplex (Vermeltfoort and Raijmakers, 1992; Vermeltfoort and Janssen, 1996). A linear elasticanalysis is only acceptable for low stress levels. To distinguish the safe from the unsafe area, themodel should give an accurate description of reality up to failure, which if beyond the elasticphase. Therefore, a more reliable model is desired. Applying 15% reduction on the shear forceis arbitrarily and accepting the SLS when the ULS is verified is only valid in the elastic phase.

4.5. Finite element modeling of masonry

This section treats the global finite element methodology for (quasi-)brittle materials that is usedto model masonry. It describes the material models for the different failure modes, available infinite element models, developed for the analysis of this type of materials. Distinct elementmodels are not treated, although recent results are encouraging (Udec: Roberti and Spina, 2001).The analysis is limited to the finite element codes that are available at the Department of CivilEngineering during the research at hand: Diana 7.2, Atena2D (Diana, 1998; CervenkaConsulting, 2001). One of the reasons is that commercial codes are not widely spread at thisstage. Most of these codes are developed for research purposes (Mason: Lee et al., 1996, Lee,1993; Strumas: Sicilia et al., 2001) at the department at which these are developed. Secondly,focus is not on the numerical model itself. These are only one piece of the puzzle. Wheneveranother model is available it can be fit into the procedure when required (Abaqus, Adina, Algor,Ansys, Cosmos, Marc, Msc-Nastran, Microfield, Samcef, Systus: Schaerlaekens andSchueremans, 1997).

Masonry is a material which exhibits distinct directional properties due to the mortar joints whichact as planes of weakness. In general, the approach towards its numerical representation canfocus on the micro-modeling of the individual components, or the macro-modeling of masonryas a composite. Depending on the level of accuracy and the simplicity desired, it is possible touse the following strategies, Figure 5.1 (Lourenço, 1996; TC MMM N9, 2000 [TC MMM-N9:RILEM Technical Committee, Mechanical Modeling of Masonry]):• detailed micro-modeling: units and mortar joints are represented by continuum elements

whereas the unit-brick interface is represented by discontinuous elements, Figure 4.10 (a),• simplified micro-modeling (meso-model): expanded units are represented by continuum

elements whereas the behavior of the mortar joints and unit-mortar interface is lumpedin discontinuous elements. These interface elements represent the preferential cracklocations where tensile and shear cracking occur, Figure 4.10 (b),

• macro-modeling: units, mortar and unit-mortar interface are smeared out in thecontinuum, Figure 4.10 (c).

93

Interface mortar/brick

mortar brick composite“mortar”“brick”

Micro-model (a) Meso-model (b) Macro-model (c)Figure 4.10: Modeling strategies for masonry structures : (a) detailed micro-modeling, (b)simplified micro-modeling and (c) macro-modeling

One modeling strategy cannot be preferred over the other because different application fieldsexist for micro- and macro-models. Micro-modeling studies are necessary to give a betterunderstanding about te local behavior of masonry structures. This type applies notably tostructural details. Macro-models are applicable when the structure is composed of solid wallswith sufficiently large dimensions so that the stresses across or along a macro-length will beessentially uniform. Clearly, macro modeling is more practice oriented due to the reduced timeand memory requirements as well as a user-friendly mesh generation. This type of modeling ismost valuable when a compromise between accuracy and efficiency is needed (TCMM N9,2000).

Because historical buildings are considered to have large dimensions, the focus will be on themacro-modeling of masonry. The intention is not to go into the behavior of structural details,or the interaction between stone and mortar, as this is beyond the scope of this thesis. Target isto use existing models as a tool to distinct the safe from the unsafe region within the constraintof an acceptable accuracy and efficiency.

4.5.1. Macro-model - anisotropic continuum modelingIn large structures, the knowledge of the behavior of the interaction between units and jointsusually does not determine the global behavior of the structure. In this case, it is more adequateto resort to continuum models, which establish the relation between average stresses and averagestrains in masonry. Difficulties of conceiving and implementing macro-models for the analysisof masonry structures arise especially due to the fact that almost no comprehensive experimentalresults are available (either for pre- and post-peak behavior), but also due to the intrinsiccomplexity of formulation of anisotripic inelastic behavior (TC MMM N9, 2000).

Formulations of isotropic quasi-brittle materials behavior (Diana, 1998; Cervanka Consulting,2001) consider, in general, different inelastic criteria for tension, compression and shear, Section4.5.2-4.5.4.

The recently developed models (Rots, 1993a, 1993b; Lourenço and Rots, 1993; Lourenço et al.,1994; Lourenço, 1996) combine the advantage of modern plasticity concepts with a powerfulrepresentation of anisotropic material behavior, which includes different hardening/softening

94

Rankinevon Mises

σ2

ft

ft

fc

fc

σ2RankineDrucker Prager

ft

ftfc

fc

σ1σ2

τ12

Rankine type criterion

σ1

σ2

τ12

Hill type criterion

σ2

σ1

ftft

fc

fc

τ0τ1 τ2 τ3

Diana model

Figure 4.11: Rankine and Hill-type yield surfaces for masonry composite material implementedin Diana (Diana, 1999)

behavior along each material axis.

The model includes a combination of a Rankine-like yield surface for the tensile regime and aHill-like yield surface in compression.

The Rankine-like yield criterion in the principal stress space reads (Feenstra and De Borst, 1995),Figure 4.11 (upper left):

(4.33)σ σ σ σ τ1 2 1 2

2

122

2 20+ + −

� � + =

The simplest Hill-like yield surface that features different compressive strengths along thematerial axes is a rotated centered ellipsoid in the full plane stress space (σ1, σ2, τ12), Figure 4.11,(bottom left). Following expression is applied:

(4.34)A B C Dσ σ σ σ τ12

1 2 22

122 1+ + + =

were A, B, C and D are four parameters such that B²-4AC < 0, in order to ensure convexity.These four parameters can be deduced from the masonry material properties directly (Lourenço,1996).

For the Hill-type criterion, both von Mises and Drucker-Prager are implemented for the case ofcombined plasticity modeling in Diana (Diana, 1998). A similar model is available in Atena2D.(Kupher et al. 1969; Cervenka Consulting, 2001).

95

displacement

σft

Direction xDirection y

σfc

displacementFigure 4.12: Masonry behavior in uniaxial tension (left) and uniaxial compression along twoorthogonal directions (right) [according to (TC MMM N9, 2000)]

εσ

fc

ThorenveldtDiana

(multi)linearhardening/softeningDiana, Atena2D

σε

Gf,c/h

ε

Gf,c/h

fc fc

Gf,c/h

σParabolichardening/SofteningDiana

Figure 4.13: Different possibilities to model the masonry behavior in compression(Thorenveldt et al., 1987; Diana, 1998; Cervenka Consulting, 2001)

The different material behavior of masonry in uniaxial tension and uniaxial compression alongtwo orthogonal directions, is represented in Figure 4.12.

The merit of Lourenço’s model is in the anisotropic extension of the material behavior. Thecombined anisotropic material behavior is described in the elastic, plastic as well as post-peak(thus anisotropic crack-propagation). The application of the model in structural modeling leadsto excellent results, both in terms of collapse loads and in terms of reproduced behavior(Lourenço et al., 1998). Unfortunately, these models will only be available in a future releaseof Diana.

For the moment the modeling is limited to isotropic combinations of Rankine with Drucker-Prager or Rankine with von Mises yield surfaces. The anisotropic combination with the plasticor post-peak behavior is not implemented yet, nor in Diana 7.2 (Diana, 1998), nor in Atena2D1.0 (Cervenka Consulting, 2001).

4.5.2. Masonry in compressionFor masonry in compression, different models for the post-peak behavior can be used to modelthe masonry crushing. Depending on the finite element code at hand, the possibilities are shown,Figure 4.13.

From the experimental results, Chapter 5, it is demonstrated that the Thorenveldt behavior very

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ε

σft brittle (multi)linear

σft

εGf,I/h

exponentialσ

ft

ε

(a) (b) (c)

Gf,I/h

Figure 4.15: Masonry in tension - post peak behavior (Diana, 1998; CervenkaConsulting, 2001; Reinhardt, 1984; Hordijk, 1991)

ft

ftσ1

σ2

ft

ft

σ1

σ2

fc

fc

Constant tensioncutoff(Diana)

Linear tensioncutoff(Atena2D, Diana)

(a) (b)

Figure 4.14: Linear and constant tension cutoff (Diana, 1998;Cervenka Consulting, 2001)

well approximates the real (post-peak) behavior (Schueremans and Van Gemert, 2000). Thismodel is not available in Atena2D. Therefore, the linear post-peak behavior is taken as analternative, which originates from concrete crushing.

4.5.3. Masonry in tensionFor masonry, two tension cutoff criteria are available, Figure 4.14. From an experimental pointof view, neither is perfect with respect to masonry (Page, 1981, 1983; Lourenço, 1996). The firstmodel, Figure 4.14 (a) implemented in Diana is further used by Lourenço. The second model,Figure 4.14 (b), is also implemented in Atena2D. As it is extremely difficult to obtain test resultsin the area of biaxial tension-compression state, the number of test results in literature is verylimited, moreover, the scatter on available test results do not allow to prefer one model above theother (Page 1981, 1983).

Besides the tension cutoff, different possibilities apply for the post-peak behavior, Figure 4.15.

No direct tensile tests on masonry are performed. Extensive research of masonry in tensionregime (Van der Pluijm, 1999) shows that the exponential post-peak behavior very well simulatesthe real behavior (Cornelissenet al., 1986; Hordijk, 1991; Reinhardt, 1984; Van der Pluijm,1999).

4.5.4. Masonry in shearFor the post-peak behavior of masonry in shear, a shear retention factor (rg) is defined (Diana,1998, Cervenka Consulting, 2001; CUR 171, 1994; Van der Pluijm, 1999):

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τ

c σ>0

σ=0Gf,II/h

τ

γ

rg εnn

γ

rg.G

εnn

(b)(a) (c)

γ

Constant shearretention rg

Variable shearretention rg1

Figure 4.16: Masonry shear behavior and shear retention curves (Diana, 1998;Cervenka Consulting, 2001)

(4.35)τ γ= r Gg

For the shear retention factor (rg), different options are available in both finite element codes.The outcome can be set constant between [0,1] or variable, with relation to the vertical dilatation(εnn), Figure 4.16 (a and b). Target is to have a closer overlap with the reduction in shear capacityas found in the experimental results, Figure 4.16 (c) (Lourenço, 1996).

4.6. Material properties

The former sections gave a general description of material models for masonry. For practicalapplication of these models, several material parameters are required. In Chapter 5, thesematerial parameters are focused on. Attention is not only on the mean value, but also on thespread of the results. Elastic as well as plastic and post-peak behavior are looked at. This isnecessary to account for stress-redistribution after the peak load. This, for example, is importantfor shear walls, Application 4, Chapter 6. After preliminary tensile cracking, a stress-redistribution takes place. A compression band occurs and the wall fails due to shear in a largeshear-band area.

In general, following material parameters are treated:• Elastic properties: Young’s modulus (E), Shear modulus (G) and Poisson’s ratio (ν),• masonry in compression: compressive strength (fc), fracture energy (Gfc),• masonry in shear: friction angle (tan(φ)), cohesion (c), fracture energy (Gf,II),• masonry in tension: tensile strength (ft), fracture energy (Gf,I).

Furthermore, Chapter 5 will treat triaxial testing of masonry cores, drilled from wallets. One ofthe aims of these tests is to check whether or not the anisotropic behavior is considerable. In casenot, the use of isotropic models may be justified. If it is, future research should put moreemphasis on the anisotropic modeling of masonry.

4.7 Conclusions

This chapter gave an overview of the masonry material models that will be used for the limit stateformulations in the different applications, Chapter 6. Analytical as well as numerical, finite

98

element models are treated. For the analytical models, focus is on masonry in compressiveregime. Because centric compression will hardly ever be the case, eccentric loading is treatedin detail. Besides non tension material model (NTM), the influence of a limited tensile strengthand plastic behavior are treated. Arching is described subsequently. For both, the design modelsavailable in EC6 are mentioned. For the latter, some caution is needed before using these modelsfor the limit state formulation in a probabilistic approach in order not to obtain biased results.

Analytical models for lateral loading and shear walls are rudimentary. They demonstrate theneed for numerical models, to obtain greater similarity with reality.

For the numerical modeling of masonry, attention is paid to finite element modeling. Althougha detailed micro-model would deliver more accurate results, a macro-model is used for reasonsof efficiency. The global anisotropic continuum model in the elastic, plastic and post-peakphase, proposed by Lourenço, is treated. The limitations with respect to available finite elementcodes - Diana 7.2 and Atena2D - are listed.

From these models, a variety of material parameters are derived. When a probabilisticevaluation procedure is aimed at, the distribution of these parameters need to be determined.This topic is addressed in the subsequent chapter.

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5 Experimental research

5.1 Introduction

Strength and stiffness of masonry are the material properties used in the design or evaluation ofmasonry structures. In a partial safety factor method (level I method), use is made of a masonrypartial safety factor (γm) (EC1, 1994) to include the spread or physical uncertainty on the materialcharacteristics.

When a probabilistic evaluation method is used, the material properties are defined as randomvariables. This is done to examine the effect of the material properties uncertainty on the failureprobability of the whole structure. Part of the uncertainty is inherent to the material characteristicitself, part of it is inherent to the test method used, the limited number of samples taken or therestricted knowledge of the phenomenon (Ditlevsen, 1982). Using masonry, additionaluncertainties are traced when compared to steel or wood. Masonry is a composite material. Itis built from bricks and mortar following a certain regular framework. Relatively large testsamples are needed to retrieve the material properties of the composite material.

Dealing with existing buildings, there are different sources to acquire the material propertiesneeded for a safety assessment (Schaerlaekens et al., 1999). The following are possible sourceslisted in the order of decreasing reliability but in the order of increasing simplicity for gatheringlarge numbers of data, required for statistical purposes.

The most feasible samples are those removed from the building itself. Direct testing ofspecimens from the building, include effects such as workmanship, actual three dimensionallayout, original materials and environmental influences (NBS 62, 1977). In case the samples aresufficiently large, they would represent the ideal samples and would give ideal estimates for themasonry material properties, although the reliability of these results will depend upon the methodemployed in the removal of samples and the implementation of the appropriate test method.Although these would be ideal samples, it is a very time-consuming, expensive procedure andtherefore not optimal for statistical processing that requires a relative large number of samples.For historical monuments this type of heavily destructive tests is not acceptable (Charter ofVenice, 1964; Charter of Krakow, 2000).

To overcome the problem of taking samples from the building itself, masonry samples - smallor large wallets - are rebuilt with original/new bricks and a similar fresh mortar based on thechemical analysis of the original mortar (Pistone and Riccati, 1988, 1991; Binda et al., 1988).The effect of time and original workmanship are lost, but they are the best alternative as still themasonry properties are directly derived on experimental basis. Due to the large scale, mostresearches are restricted to 3 full scale walls. This is mathematically the minimum required forstatistical analysis.

Many test methods have been developed to retrieve the material properties of the componentsbrick and mortar and in addition, numerical relationships between component characteristics andcomposite characteristics have been determined (Binda et al., 1988; TNO report BI-78-44, 1978;

100

Lourenço, 1996). Because of the numerical model involved, an extra model uncertainty has tobe taken into account.

Tests on bricks and mortar are less time-consuming, less expensive and can be performed inrelatively large numbers which enables the probability distribution function of the materialproperties to be determined.

With regard to the probabilistic evaluation method, the probability distribution function of themasonry is looked for. Therefore, the question arises whether the existing or proposed analyticalrelationships between components and composite are valid for a wide range of values, so it issuitable for statistical processing.

Finally, literature becomes an additional source of experimental data that is gaining field. Manydifferent test setups and results can be retrieved as listed in several reviews (CUR 171, 1994;Hordijk, 1991; NBS 62, 1977; TNO report BI-78-44, 1978). They classically deal with masonryin a particular load regime such as compression, tension, shear, bending and biaxial or triaxialstress regime (Hendry, 1998; Lenczer, 1972; Page, 1981; Sahlin, 1971; Vintzileou and Tassios,1995). One of the main conclusions of these literature reviews is that as a consequence, seldomall data of a particular type of masonry, required in a finite element analysis, can be gathered.On the other hand, the need for experiments and experimental data grow with the increasinginsight in the behavior of masonry and the needs coming from more accurate numerical modelingsuch as non-linear finite element methods and fracture mechanics.

Recent research programs try to meet this lack of consistent results (CUR 171, 1994; Hordijk,1991; Lourenço, 1996; Van der Pluijm and Vermeltfoort, 1991; Van der Pluijm, 1992, 1996;Vermeltfoort, 1995; Vermeltfoort and al., 1993; Vermeltfoort and Janssens, 1996; Vermeltfoortand Rijmakers, 1993). One of the targets of these research programs is to obtain practical designrules for masonry based on an experimental/numerical basis (CUR 171,1994). An experimentalresearch program was set up in which the material properties of masonry are gathered to meetthe requirements of numerical models.

In case a probabilistic evaluation method is used for an assessment of masonry structures, theneeds are similar. In addition, the number of test samples should be increased in order toestimate the probability distribution functions that are representative for the uncertainty on thematerial properties. Moreover, evaluation deals with existing structures, not with design of a newmasonry structure, adding a major difficulty in gathering consistent information.

Parallel to the evolution in design methods (from level 0: elastic methods tot level IIIprobabilistic methods) and modeling (from elastic analysis to nonlinear finite element analysisand fracture mechanics), there is a tendency towards reporting all individual results (Van derPluijm, 1999). In the past usually no more than the mean value was used and thus reported. Forretrieving characteristic values of material properties for design purposes using a partial safetymethod, the scatter is required. Therefore, in many cases, the coefficient of variation is listed too,sometimes even the number of samples between brackets. In publishing mean values andcoefficients of variation only, a lot of information is lost. To enable the use of probabilisticmethods, the individual data are required (or their probability density function). Therefore, thetendency towards publishing individual test results is appreciated. The individual test results of

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this research program are listed in Annex B.

In the future this should lead to extensive and detailed databases on which a reliability analysiscan be based. In case of existing buildings it should be possible to gather data from comparableconstructions. The international community is aware of the lack of consistent data on existingmasonry structures and the fact that for the moment probabilistic approaches usually fail fromlack of sufficient data (EC1, 1994). To meet this requirement, data are collected on aninternational basis using a database structure that can be consulted via the internet (Pande, 1995).These should become a full alternative for in site testing (NBS 62, 1977). For the moment,available test data on historical masonry are too limited to be useful for this study.

This chapter deals with the experimental tests and their statistical processing. Main objective isto obtain the probability distribution of the main material properties of masonry needed in a finiteelement code (Atena2D, Diana) to evaluate the safety of the structure. Therefore, research isfocused on three levels: the components brick and mortar, the composite masonry and therelationship between components and composite. An answer to the following questions is lookedfor:• what is the variability of the composite masonry (distribution type and parameters),• can the material properties of masonry be estimated from the components brick and

mortar using existing test methods and numerical relationships,• are these test methods adequate to gather this information in site from an existing

masonry structure ?

As the number of test results is relatively high for specific test setups, some interesting remarksregarding size-effect, different test methods or anisotropy can be made.

Strength and stiffness of masonry in compressive regime are the main material properties thatwill be dealt with. These are chosen as masonry and certainly historical masonry, is mainlydesigned to act in compression. On the other hand failure is mainly due to cracks caused bytensile or shear stresses as masonry is a quasi-brittle material with a limited strength in tensionand shear (Lourenço, 1996). Because of its quasi-brittle behavior and its limited strength intension and shear, the stress-strain relationships in tension or shear are not examined as often asthe compressive regime. Recently, research in this field gains interest. Complex test setups arebuilt to obtain reliable test results (Van der Pluijm, 1999; Vermeltfoort and Janssen, 1996). Thegoal of this work is not of experimental nature solely. Therefore, when needed and available, thevalues for the tensile and shear regime are supplemented with values taken from literature (CUR171, 1994; Schaerlaekens and Schueremans, 1997). In this work only a limited amount of tensileand shear tests are performed. Additionally, masonry does not always act in a uni-axial stressstate. To gather information about masonry in a multi-axial stress state, a triaxial testing devicewas acquired by the Reyntjens Laboratory. Test setup and results are discussed.

To enable comparison of test results, only one type of bricks and one type of mortar is used. Theresearch is limited to a single bond: the Flemish bond. The number of test samples is takensufficiently high to decrease the statistical uncertainty (Schaerlaekens and Schueremans, 1997)and to enable mutual significant comparisons. Different test methods are used to check theiradequacy. By using a cement-lime mortar, an interesting overlap with the doctoral research ofRoald Hayen is established (Hayen, 1999). Besides, many historical masonry buildings are

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constructed with a cement-lime mortar (Van Balen, 1991; Van de Vijver, 2000).

The individual test results can be retraced in Annex B. The following sections contain theinformation needed to elaborate the applications, discussed in Chapter 6. Section 5.2 deals withthe behavior of the components mortar and brick. Section 5.3 treats the composite masonry.After comparative tests, the numerical model proposed in Eurocode 6 is extended to randomvariables, Section 5.4. The probability distribution of the compressive strength of masonry issimulated based on the compressive strength of bricks and mortar. Homogenization techniquesare used to calculate the probability distribution of the stiffness of masonry based on thecomponents probability distribution. This is elaborated in Section 5.5. Section 5.6 handlesmasonry in shear regime. Section 5.7 treats the tensile regime. Masonry in a multi-axial stressstate is described in Section 5.8. Test setup and results are discussed. The global results andmain findings are summarized in Section 5.9. The tests on masonry executed in the frameworkof this study, are summarized in Table 5.1.

Type of testsample

test geometry Data-acquisition [number ofsamples]

mortar standardmortar beams

3-point bending test compressive test

l×h×w = 160×40×40 mml×h×w = 40×40×40 mm

ft [53]fc, σ-ε, E [108]

cores triaxial test α = 0q,� = 150 mm, h = 300 mm yield criterion[28]

brick couplets compressive test 2-layers of brick and a single bedjoint, l×h×d = 188×126×48mm

fc, σ-ε, E [50]

cores compressive test vertically drilled from bricks�=50 mm, h =58 mm

fc, σ-ε, E [51]

prisms compressive test b×h×d = 160×45×25 mm fc, σ-ε, E [40]

masonry cores compressive testshear test

α = 0q, 45q� = 150 mm, h = 300 mm

fc, σ-ε, E, Gfc [5] τ-γ, GfII [5]

tensile test vertically drilled from couplets� = 50 mm, h = 126 mm

ft [56]

triaxial test α = 0q, 45q� = 150 mm, h = 300 mm

yield criterion[30,15]

masonrypillars

compressive test 6-layers of bricks (cross bond)w×h×d = 188×338×188 mm

fc, σ-ε, E[20]

wallets compressive testshear test

α = 0q, 45qw×h×d = 582×570×188 mm

fc, σ-ε, E, Gfc [3] τ-γ, GfII [3]

Table 5.1: Overview of experimental research

5.2 Components - brick and mortar

103

l = 188 mm

h = 48 mm

w = 88 mm

Figure 5.1: Hand made facing brick Kempenbrand

X

X X X

YYY

Y

Y

X

h = 16

0 m

m

100

mm

Figure 5.2: Different types of test samples for bricks

5.2.1 BricksThe used brick is a hand made facing brick module 50, according to the Belgian Standard NBNB23-002 (1986), with a size of: length×width×height = 188×88×48 mm, type Kempenbrand,Figure 5.1. This molded hand made brick has a rough outlook and surface. It is baked in a ringoven using carbon on a temperature of 1100bC. The stone is produced from pure Kempish oldquaternary clay that originates from the Waliaan interglacial. The color of the brick is light tilldark red and it shows light till dark purple-brown burned spots. They contain a groove on oneside (Terca, 1999). A molded hand made brick has been chosen as its compressive strength israther limited, what can be expected in case an assessment of an historical masonry structuresis appealed for. As the confining pressure of the triaxial cell is limited to 17 MPa, the limitedcompressive strength is an advantage too. Finally, the same brick was already used in a formerresearch program (Schaerlaekens and Schueremans, 1997), adding extra test results and referencevalues.

As for bricks, compressive tests are performed on three types of test samples: cores ‰50mm witha height of 44 mm, couplets with a height of 120 mm and prisms sawn from bricks with a heightof 160 mm, Figure 5.2. These three types of specimens are chosen as they limit the degree ofdestruction in case of an existing building.

To produce the different test samples, the required bricks are taken randomly from a totalpopulation of 2400 bricks (3 pallets of 800 bricks each).

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5.2.2 Cores ‰‰‰‰50mm with a height of 44 mmThe first type of test samples are cores ‰50 mm with a height of 44 mm to meet the correct ratiobetween height (h2) and surface (S) according to the Belgian Standard NBN B15-220 (1990):

(5.1)hS

2

1=

As the bricks contain a groove, this groove is filled with a standard mortar first, according toPrEN 772-1 (1999). After hardening, 3 cores ‰50 mm are drilled from each brick. They arerectified until a height of 44 mm is reached. The velocity of the displacement controlledcompressive test yields: v = 1 mm/min. The same velocity is taken for the other two types ofbrick samples. The results are summarized in Table 5.2. The statistical plots are shown inFigures 5.3-5.5. The coefficient of variation equals 37%. This is almost completely the resultof a variation inherent to the material property itself. The coefficient of variation on thegeometry and density are of the order of magnitude of 1 till 4%, see Annex B.1.

Cores ‰50 mm with a height of 44 mm fc

sample mean: [N/mm2]xsample spread: s [N/mm2]coefficient of variation: cov [%]number of test samples: nestimated probability distribution function: PDF

6.342.373751lognormal (LN) - Pareto (GPD)

Table 5.2: Cores ‰50 - statistical summary of results

Although many authors suggest a lognormal (LN) probability distribution function for this typeof material properties (Melchers, 1999; CUR 171, 1994), relatively heavy tails are clearly visible,Figure 5.3. These suggest an extreme value distribution type. As the number of observations issufficiently large, an extreme value analysis is performed. A complete overview on extremevalue analysis is beyond the scope of this study and can be found elsewhere (Caers, 1996;Beirlant et al., 1996; Maes and Caers, 1998; Vynckier, 1996). For a very comprehensivesummary for practical use on this topic, the reader is referred to Willems (Willems, 2000).

Based on an extreme value analysis, it is seen that the data best fit a Generalized ParetoDistribution (GPD):

(5.2)( )G x x xth= − + −� �

−

1 11

ης

η

The extreme value index η defines the shape of the tail. For increasing η, the tail becomesheavier. A new type of QQ-plot was introduced by Beirlant (Beirlant et al., 1996) to estimate ina visual way the extreme value index and the corresponding distribution class. It is referred toas the generalized quantile plot, or UH-plot, Figure 5.4. In Figure 5.4, left axis, the UH-estimation is shown as a function of the number th of extremes that are used in the estimation.A Hill-type estimator is used. On the right axis, also the mean squared error (MSE) of the

105

Figure 5.3: Histogram, estimated probability distribution function and quantile-quantile-plots(QQ-plots)

weighted linear regression in the UH-plot is shown as a function of th. For th<30 the extremevalue index fluctuates around η = 0.25. The same is concluded from the Pareto QQ-plot, Figure5.5 and the Hill estimation of η, Figure 5.6. Except the last value, the points tend to show alinear trend. This is rather strange as it is a distribution type with very heavy tails, which is quitesurprising for a material characteristic such as the brick compressive strength. The optimaltreshold th = 30 is determined that minimizes the mean squared error (MSE). At this threshold,the observation xth = 5.73 N/mm2 is found. The parameter ζ = xth× η = 5.73 × 0.25 = 1.43 N/mm2

(Willems, 2000).

106

-0,5

-0,4

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0,4

0,5

0 10 20 30 40 50number of observations above treshold

η=

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

Figure 5.4: Cores ‰50 compressive strength fc. Left axis (´): UH-estimation of extreme valueindex η, right axis (?): mean squared error of weighted linear regression in the UH-plot.

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6( -ln( 1-G(x) ) )

Figure 5.5: cores ‰50 - compressive strength fc. Pareto QQ-plot.

107

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

0 10 20 30 40 50number of observations above treshold

η

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

Figure 5.6: Cores ‰50 compressive strength fc. Left axis (´): Hill-estimation of extreme valueindex η, right axis (?): mean squared error of weighted linear regression in the Pareto QQ-plot.

It is believed that the heavy tails are caused by the mortar filling of the groove in the brick. Astandard mortar according to prENV 772-1 (1999) was used for that. Because of this mortarfilling, a composite material is tested, instead of the pure brick material. This standard mortaris partly present in some of the samples. It has a relatively high strength, compared to the brickmaterial, which causes the extremes as they are observed in the test results. Therefore alognormal distribution type will be adopted in the further analyses, as it limits the tail behavior,that can not be counted on in reality.

5.2.3 Couplets with a height of 120 mmThe second type of test samples are couplets with a height of 120 mm according to the BelgianStandard NBN B24-201 (1974). Seen the length of the bricks is smaller than 400 mm and theratio height on width is smaller than 0.55, the test samples consist out of two bricks placed ontop of each other, joined with a mortar bed joint. Considering the irregular shape, the testsamples have been rectified with a standard mortar layer on top and bottom, Figure 5.2. Theresults are summarized in Table 5.3. The compressive strength (fc), the Young’s modulus (E)based on the compressive plate displacement and the fracture energy (Gfc) are summarized.

The fracture energy (Gfc) is calculated as the area beneath the stress-displacement relationship,starting from the strain at the peak stress and above the residual stress (σr K1MPa) at the end ofthe compressive test (Lourenço, 1996). To obtain consistent results between the different heightsof the sample types, the values are scaled to a reference length lref = 200 mm, see Annex B. Toreach the residual stress, the total displacement of the compressive plate was taken large enough.A total strain of approximately 0.015 mm/mm was required for that purpose Figures 5.8, 5.9 and5.10.

108

Couplets with a height of 120 mm fc,y [N/mm2] Ey[N/mm2]

Gfc [Nmm/mm2]

sample mean: xsample spread: scoefficient of variation: cov [%]number of test samples: nprobability distribution function: PDF

5.161.563050LN

3811092950LN

2.280.954150LN

Table 5.3: Couplets - statistical summary of the material properties

In contrast with the former type of test samples, a lognormal distribution type very well fits theobserved test results. The tails are less heavy, see QQ-plots, Annex B.2.

The Young’s modulus is determined in the elastic area between the stress levels 0.5 N/mm2 and2/3fc. With a sample mean that equals 381 N/mm2 a non-realistic low value is obtained.Although these tests were performed on the Dartec test bank (Dartec, 1996), the stiffness of thetest bank and more particular the effect of the ball-and-socket-joint can not be neglected at all.This is mainly due to the initial compression of grease between the ball-and-socket-joint. Fromthe test results on masonry prisms, cores and wallets the influence of the test bank stiffness wasestimated using deformation measurements based on linear varying deformation transducers(LVDT). A linear regression and variance analysis is used to model the influence of the test bankstiffness on the measured Young’s moduli. This correction function will be treated shortly inSection 5.3.4. Because the correlation is rather small - the compression of the grease in the ball-and-socket-joint is a relatively random process - a substantial model uncertainty is introduced.This makes the established regression function not very attractive for practical use and stimulatesthe calculation of the Young’s modulus based on direct measurement of deformations usingLVDT.

It is noticed that the coefficient of variation on the Young’s modulus is very well the same as thecoefficient of variation on the compressive strength, Table 5.3. The same conclusion can bedrawn from the other tests as reported in the following sections.

The positive correlation between the different material properties (E, fc, Gfc) is represented inTable 5.4. The correlation between the compressive strength and the original Young’s modulusamounts 83%. This value is higher than might be expected based on former research (Claes andHermans, 1999: ρ(E,fc) K 64 %). There is also a correlation between the compressive strengthand the fracture energy (ρ(fc, Gfc) = 51 %) and between the Young’s modulus and the fractureenergy (ρ(E, Gfc) = 37 %), although the latter one is rather limited.

correlation (ρ) fc [N/mm2] E [N/mm2] Gfc [Nmm/mm2]

fc [N/mm2] 1

E [N/mm2] 0.83 1

Gfc [Nmm/mm2] 0.51 0.37 1Table 5.4: Correlation between the material properties

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The individual results, the stress-strain relationship as well as the detailed sample statistics canbe found in Annex B.2.

5.2.4 Prisms sawn from bricks with a height of 160 mmAs the stiffness values of the bricks based on the plate displacement show a great distortion andcorrection for the plate displacement introduces extra uncertainty on the measurements,additional tests on prisms sawn from bricks with a height of 160 mm, Figure 5.2, are performed.These samples are tested along their length axis (x-axis) so linear varying deformationtransducers can be placed on a reference length (lref) of 100 mm. The test conditions are identicalto the former compressive tests, although a different test bank was used (Schenck, 1970). Theresults are summarized in Table 5.5.

Prisms sawn from bricks with a height of 160 mm fc,x [N/mm2] Ex [N/mm2]

sample mean: xsample spread: scoefficient of variation: cov [%]number of test samples: nestimated probability distribution function: PDF

8.001.932440LN

17114002338LN

Table 5.5: Prisms - statistical summary of the material properties

Again, for more detailed information on the individual results and sample statistics, the readeris referred to Annex B.3. The sample mean of the Young’s modulus equals: Ex = 1711 N/mm2.This value will be used in further applications. The value of the compressive strength that wasdetermined is higher than in the former tests, although this is the only test in which pure brickmaterial is tested. Remark that again, the coefficient of variation for both material properties isaround 25%.

5.2.5 Discussion of results - compressive strength fcThree different tests were used to gather information about the compressive regime of one typeof masonry bricks. As for the compressive strength, different results are obtained. Whether ornot these differences are significant will be analyzed in this section. The results on thecompressive strength are summarized in Table 5.6. Subsequently are given: mean value, spread,coefficient of variation, number of samples and type of probability distribution.

Compressive strength fc [N/mm2] cores ‰50,h = 44 mm

couplets, h = 120 mm

prisms, h = 160 mm

Sample mean: [N/mm2]xSample spread: s [N/mm2]Coefficient of variation: cov [%]Number of samples: nprobability distribution function: PDF

6.342.3737.451GPD/LN

5.161.563050LN

8.001.9324.240LN

Table 5.6: Compressive tests on bricks

The technique of hypotheses tests (Van Dyck and Beirlant, 1999; Wackerly et al., 1996) is usedto check the significance of these results. To obtain a sufficient level of evidence, a significance

110

level of 95% is preset (α = 0.05). The null hypothesis H0 and the alternative upper tail hypothesisH1 read:

(5.3)HH

0 1 2

1 1 2

::µ µµ µ=>

The test statistic (T) for the difference in mean value of the population, with unknown spread (σ1and σ2 not known) but σ1 = σ2 = σ3:

(5.4)T X X

Sn np

= −

+

1 2

1 2

1 1

in which Sp is the pooled estimator that equals:

(5.5)( ) ( )

Sn S n S

n np2 1 1

22 2

2

1 2

1 12

=− + −

+ −

The null hypothesis is rejected - interpreted as proof or strong conclusion - in case:

(5.6)t to n n> + −α , 1 2 2

The p-value or attained significance level is the smallest level of significance for which theobserved data indicate that the null hypothesis should be rejected, is calculated using:

(5.7)P F t in case HTn n= − >

+ −1

1 2 2 0 1 1 2( ) :µ µ

In this way a type I error (made if H0 is rejected when H0 is true) is controlled for.

To limit the chance on a type II error (made if H0 is accepted when H1 is true, for a difference inmean values equal to: µ1-µ0 = 1 N/mm2) to a level of β=0.05, the required number of samples nβcan be derived from operating characteristic plots (OC)-plots (Van Dyck and Beirlant, 1995).The results of a mutual comparison of the test results above, are given in Table 5.7.

Table 5.7 indicates indeed that there is a significant difference in average compressive strengthbetween cores ‰50, couplets and prisms. With a preset confidence level of β=0.05 for a type IIerror as described above, the required amount of test samples varies between nβ= 32 - 52. Onlyin the case of prisms this condition is not met.

Hypothesistest

cores ‰50, h = 44 mmversuscouplets, h = 120 mm

couplets, h = 120 mmversusprisms, h = 160 mm

prisms, h = 160 mmversuscores ‰50, h = 44 mm

111

Sp 2.01 1.72 2.19

t0 2.95 7.78 3.59

tα 1.660 1.662 1.662

Conclusion Reject H0, strongconclusion

Reject H0, strongconclusion

Reject H0, strongconclusion

p-value 0.002 - low: verysignificant

6.5 10-12 - low: verysignificant

2.7 10-4 - low: verysignificant

nβ 44 32 52Table 5.7: Summary of hypotheses tests

The explanation for these differences should be looked for in two directions. Besides the effectof the mortar filling of the groove in the bricks, for cores ‰50 with a height 44 mm and coupletswith a height of 120 mm, the difference is merely caused by the size effect of the samples.Eurocode 6 provides a normalizing factor to correct for the different sizes of test samples (EC6,1995):

fb = δ×fb,m = 0.82×6.34 = 5.20 N/mm2 for cores ‰50,fb = δ×fb,m = 1.11×5.16 = 5.73 N/mm2 for couplets with height 120 mm.

The values are normalized to a standard sample size of 100×100 mm. Whether or not this is arepresentative size for bricks is questionnable (Lourenço, 1996). The difference between thenormalized sample means is no longer significant in case of 50 samples:

t0 = 1.45 < tα = 1.66. (5.8)

For comparison with the prisms, height 160 mm, sawn from the bricks, the influence of theorientation in which the compressive force is applied, plays a role. These prisms are compressedaccording to the length axis of the brick (x-axis), Figure 5.2. The cores ‰50 with a height of 44mm and couplets with a height of 120 mm, are compressed according the height axis of the brick(y-axis), similar to the masonry wallets and cores further on in this research. The brick, typeKempenbrand, is a hand made brick. Putting the clay in the mold and pressing it from top,causes a certain layering of the material which remains visual after the brick is baked. Thislayered effect leads to a significant anisotropy in the material properties, that is evidenced fromthe hypothesis tests. So, one can state that:

(5.9)[ ] [ ]E f E fb x b y, ,>

Seen the correlation between the material properties in one direction, Table 5.5, the anisotropythat is recorded on the compressive strength will more than likely appear on the other materialproperties too. In case of Young’s modulus a sufficient height of test samples is required to beable to place linear varying deformation transducers (LVDT) and obtain results with a sufficient

112

Figure 5.7: The Saint-Astier limestone quarry

accuracy. The cores ‰50 with a height of 44 mm and couplets with height 120 mm were notsuitable for that purpose. Therefore, the prisms with a height of 160 mm were added to theresearch program. Later on, using homogenization techniques to determine the Young’s modulusof the masonry based on the values of the bricks and mortar, abstraction is made from the brickbeing not isotropic.

5.2.6 MortarThe mortar is a hybrid mortar type M4, according to the Belgian Standard NBN B24-301 (1980).Following composition was used:• 158 kg/m3 cement CEM I 42.5 (1 volume part),• 103 kg/m3 hydraulic lime (1 volume part),• 1217 kg/m3 Zutendaal sand (6 volume parts) and• 292 kg/m3 water.

Water is added to obtain a uniform work ability between 1.9 and 2.1 according to NBN B14-207(1983) (Schaerlaekens and Schueremans, 1997). The hydraulic lime, type NHL 3.5 accordingto the French Standard NFP 15-311 (1996) and European Standard ENV 459-1 (1995), originatesfrom Saint-Astier (Dordogne, Périgord) at the South-West of France, Figure 5.7. The meancompressive strength fc=8.0 N/mm2 after 28 days and mean tensile strength ft = 2.0 N/mm2,according to ENV 459-2 (1994) (CESA, 1999).

The mortar samples, standard mortar beams with dimensions: length×width×height =160×40×40 mm3, are made according to the Belgian Standard NBN B14-209 (1983) and testedaccording to Belgian Standard NBN B14-208 (1969). The samples are made during theconstruction of the masonry wallets that are discussed in Section 5.3.3. Thus, the compositionof the mortar beams is representative for the masonry samples further on. They were stored inthe same laboratory circumstances as the masonry samples: 20bC and 65%RH, during 195 days.

The results of the compressive and bending tests are summarized in Table 5.8. During thecompressive tests, the compressive plate displacements have been recorded (Shenck test bankdevice). The modulus of elasticity is calculated between stress levels 0.5 N/mm2 and 5 N/mm2,which is approximately equal to 2/3fc. To account for the test device stiffness, the platedisplacement was recorded using an identical test setup but replacing the deformable mortar

113

sample with a quasi non-deformable steel sample. The deformability of the testing device isalmost linear within a range of 0-20 kN and amounts: 0.040 mm/kN. Based on this correctionmodel, the values of the Young’s modulus are adapted (Ec).

mortar samples - age 195 days, l×w×h = 40×40×160 mm3

fft [N/mm2] fc,y [N/mm2] E (Ec) [N/mm2]

xscov [%]nPDF

2.600.491953trunc. N/LN

8.311.8021.6108trunc. N/LN

410 (1398)75 (896)18 (64)107 (107)trunc. N/(trunc. N)

Table 5.8: mortar samples - statistical summary of material properties

The mean flexural tensile strength (fft) and compressive strength (fc,y) values are similar to thevalues mentioned in the technical card, see above. As for the Young’s modulus, using a linearcorrection model increases the spread significantly. The spread is no longer representative forthe material property. Test results and sample statistics are given in Annex B.4.

Therefore, the values of the Young’s modulus are verified, using LVDT’s in the same test setupas for the brick prisms. An additional series of 20 test samples has been made and tested after160 days. These values are summarized in Table 5.9.

mortar samples 2nd batch - age 160days, l×w×h = 40×40×160 mm3

fc,x [N/mm2] fc,y [N/mm2] E [N/mm2]

xs cov [%]nPDF

3.920.581520/

4.970.721535/

13886114419/

Table 5.9: Mortar samples - 2nd batch - statistical summary of material properties

The mean value of the Young’s modulus (Ey = 1388 N/mm2) corresponds well with the correctedvalues (Ec = 1398 N/mm2) listed in Table 5.8, validating the use of a mean correction for the testbank stiffness. This is not valid for the spread. It is preferable to retain the spread of the originalvalues, as the correction results in an increase of the spread, which is no longer representativefor the spread inherent to the material property itself. This is confirmed by the comparisonbetween plate displacement and measured deformation using LVDT’s in case of cores, prismsand wallets, see Section 5.3.4. The corrected Young’s modulus listed in Table 5.8 will be usedin the applications.

For the difference in compressive strength three reasons can be put forward. First there is a smalldifference in age, although this can only cause a minor difference as the ultimate strength isachieved at approximately 90 days (Hayen 1999, Hayen et al., 2001a). Secondly the size effectand slenderness play a major role. The height ratio between both test setups amounts 4. The fact

114

that this influence can not be neglected is illustrated with the values in the mid column. Theselist the compressive strength (fc,y) of the same samples after they have withstood the firstcompressive test (fc,x). The samples that were still useable (n=35) have been tested a second timeusing a standard compressive test according to the Belgian Standard NBN B14-208 (1969).Although they contain a certain degree of damage - they already have been tested up to the peakload in another direction - a higher mean value is achieved. Finally, the second series of testsamples are made by a different person. The influence of the workmanship on the results shouldnot be neglected nor underestimated. Also in practice, different parts of masonry will havedifferent strength values as they have been constructed by different workmen.

5.3 The composite masonry - masonry in compression

Different types of test samples are used to determine the main material properties of thecomposite masonry, Section 5.3-5.8. The material properties that are examined, are the elasticproperties (E, ν, G), the compressive strength (fc) and post peak behavior (Gfc), the tensilestrength (ft) and post peak behavior (Gf,I), the shear behavior of masonry (c, tan(φ), Gf,II) and thebehavior of masonry in a multi-axial stress state.

The first sections deal with masonry in compressive regime, Section 5.3-5.5. Based on differenttests, the main masonry material properties for the compressive regime are determined. Besidesthe material properties, attention is paid to the size effect and boundary conditions of the testsamples and their effect on the uniaxial compressive strength of masonry, the ability to determinethe compressive strength of masonry from the components brick and mortar and homogenizationtechniques to determine the Young’s modulus of masonry again from bricks and mortar. Someremarks concerning test setup and comparative literature results are made.

5.3.1 Small masonry pillarsTwenty small masonry pillars have been built. They contain 6 layers of bricks. Considering therough surface of the pillars, top and bottom have been rectified using a standard mortar layer forrectification according to the Belgian Standard NBN B24 -201 (1974). The total height of thetest specimens is about 360 mm, which is believed to be sufficient to obtain representative resultsfor the masonry properties (Schaerlaekens and Schueremans, 1997). After 140 days, sufficientlyto obtain complete hardening of the hybrid mortar, these pillars have been subjected to adisplacement controlled compressive test, with a velocity v = 0.01 mm/s. The stress-strainrelationship is shown in Figure 5.8. The vertical deformation of the pillars is acquired using fourlinear varying deformation transducers (LVDT), one at each side of the pillars centrally placed,with a reference length (lref) of 200 mm. One test sample got damaged during manipulation. Thedata-acquisition of another sample was judged unreliable. The resulting material properties - fc,Ey, Gfc - are listed in Table 5.10. As for the modulus of elasticity, the values based on theLVDT’s as well as the values based on the compressive plate displacement (between brackets)are given. The influence of the test bank and measurement setup is discussed in Section 5.3.4.More detailed information on this part of the test program is stored in annex B.5.

Small masonry pillars - width×thickness×height = 188×188×360 mm, age 140 days

fc,y [N/mm2]

Ey,LVDT (Ey,plate) [N/mm2]

Gf,c [Nmm/mm2]

115

0

1

2

3

4

5

6

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8

0.00

9

0.01

0.01

1

0.01

2

0.01

3

0.01

4

0.01

5

Strain ε [mm/mm]

Stre

ss σ

y [N

/mm

2 ]

h =

360

mm

w = 188 mm d = 18

8 mm20

0 m

m

LVDT

Figure 5.8: Stress-strain relationship - masonry pillars

sample mean: xsample spread: scoefficient of variation: cov [%]number of test samples: nprobability distribution function: PDF

4.260.831919(N)/LN

1673 (837)497 (181)30 (22)18 (19)N (N)

1.710.563318N

Table 5.10: Small masonry pillars - statistical summary of material parameters

Based on the histogram and QQ-plots, see annex B.5, a normal as well as a lognormaldistribution type fit equally well the limited number of observations. As the compressivestrength is a non-negative material property, a lognormal distribution type is preferred (Melchers,1999). The mean compressive strength is lower than the values found on the brick and mortarsamples. The values of the fracture energy coincide well with values found in literature (Van derPluijm, 1999; Lourenço, 1996).

Table 5.11 lists the correlation between the above mentioned material properties. As the numberof samples is rather small, these only have an indicative value. In all cases a positive correlationis retrieved. The correlation between the compressive strength (fc) and Young’s modulus (Ey)corresponds well with literature (Cur 171, 1994: ρ(E,fc) K 80 %).

correlation (ρ) fc [N/mm2] Ey,LVDT (Ey,plate) [N/mm2] Gfc [Nmm/mm2]

fc [N/mm2] 1

116

Ey,LVDT (Ey,plate) [N/mm2] 0.55 (0.72) 1

Gfc [Nmm/mm2] 0.66 0.16 (0.49) 1Table 5.11: Correlation between masonry material properties in compressive regime

5.3.2 Masonry CoresThe second type of test samples that has been used to derive the masonry material properties arecores drilled from wallets. The wallets were constructed similar with and at the same time as thewallets used for compressive tests, see Section 5.3.3. The cores have a nominal diameter of 150mm and a height of 300 mm to obtain a ratio between height and diameter that amounts 2. Fivemasonry cores have been subjected to a uni-axial displacement controlled compressive test.Remark that these results will be added to the σ1-σ2=σ3-stress plane on the σ1-axes as theconfining pressure (σ3) equals zero, Section 5.8.4. The deformations are acquired using twolinear varying deformation transducers, with a reference length lref equal to 200 mm. Theresulting material properties - fc, E, Gfc - are listed in Table 5.12. More detailed information onthis part of the test program is given in annex B.6. Remark that the residual strength (σr) equalsalmost 1 N/mm2. Although the scatter is wider, a similar residual strength can be noticed basedon the compressive tests on small pillars, Figure 5.9. Comparing the stress-strain relationshipbetween pillars and cores a difference in post peak behavior is noticed. The prisms seem to beless brittle. For the cores, after peak load a relatively sudden fall in stress is recorded. In caseof pillars the post peak strength decays more gradually. This is translated in slightly highervalues for the fracture energy (Gfc).

Masonry cores h×d = 300×150 mm - age 152days

fc,y [N/mm2]

Ey,LVDT (Ey,plate) [N/mm2]

Gf,c [Nmm/mm2]

sample mean: xsample spread: scoefficient of variation: cov [%]number of test samples: nprobability distribution function: PDF

4.540.77175/

1690 (798)684 (179)41 (22)5 (5)/ (/)

1.351.02765/

Table 5.12: Masonry cores - statistical summary of material parameters

Overall, the mean compressive strength (fc) is very similar to the value of the masonry pillars,Table 5.10. The same can be said of the Young’s modulus (E) and even of the fracture energy(Gfc). The amount of test samples is too small to retrieve a distribution type or correlation valuethat can be relied on.

117

0

1

2

3

4

5

6

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8

0.00

9

0.01

0.01

1

0.01

2

0.01

3

0.01

4

0.01

5

Strain ε [mm/mm]

Stre

ss σ

y [N

/mm

2 ]

200

mm

h =

300

mm

φ = 150 mm

LVDT

Figure 5.9: Stress-strain relationship masonry cores

5.3.3 Masonry WalletsThree small masonry wallets (V1-V3), Figure 5.10, with nominal sizes: width×depth×height =600×188× 600 mm - Flemish bond - are tested according to the Belgian Standard NBN B24-211(1978). The vertical displacements are recorded using four vertical linear varying deformationtransducers, with a reference length (lref) of 460 mm. The horizontal displacements are recordedusing six LVDT’s. One at front and back (lref = 400 mm) and two at each side (lref = 100 mm).The results of the displacement controlled compressive tests are summarized in Table 5.13. Thevertical stress-strain relationship is shown in Figure 5.10.

masonry wallets, age 141 days w×d×h = 577×188×570 mm

Wallet V1

Wallet V2

Wallet V3

s covx[%]

fc [N/mm2] 5.81 5.68 6.72 6.07 0.57 9.3

Ey,LVDT(Ey,plate) [N/mm2]

1670(1200)

1585(1018)

1672(1156)

1642(1124)

50(94)

3(8.5)

νxy 0.28 0.13 0.13 0.19 0.06 31

νyz 0.15 0.20 0.25

Gfc [Nmm/mm2] 2.83 3.26 2.09 2.73 0.60 22Table 5.13: Masonry wallets - statistical summary of material parameters

118

0

1

2

3

4

5

6

7

8

0

0.00

1

0.00

2

0.00

3

0.00

4

0.00

5

0.00

6

0.00

7

0.00

8

0.00

9

0.01

0.01

1

0.01

2

0.01

3

0.01

4

0.01

5

strain ε [mm/mm]

Stre

ngth

σy [

N/m

m2 ]

h =

570

mm

w = 577 mmd =

188 m

m

lref = 400 mm

l ref =

460

mm

lref = 100 mm

Figure 5.10: Masonry wallets - vertical stress-strain relationship

The sample mean of the compressive strength is relatively high (fc=6.07 N/mm2). It is almostequal to that of the weakest component of the composite, namely the brick, as measured on thebrick cores ‰50 (fc=6.34 N/mm2). The spread on the compressive strength is very low. Thismight indicate that taking larger samples, a more homogenized material is obtained, resulting inlower spread on the material properties. As larger samples are taken, local effects as intrusionsor defects, no longer play a significant role. Their effect is compensated for by neighboringmaterial. This is elaborated in more detail in Section 5.5. It has to be kept in mind that onlythree samples are available, so the values of the spread are indicative values, which should notbe relied on too much. The Young’s modulus (E = 1642 N/mm2) coincides very well with thevalue measured on pillars (E = 1673 N/mm2) and masonry cores (E = 1690 N/mm2). Based onthe horizontal LVDT’s a value for the Poisson’s ratio is calculated. This again is done betweenstress levels 0.5 N/mm2 and 2/3fc. The spread is relatively large compared to the other materialproperties determined, indicating that this material property can not be measured as easily ascould be experienced in practice. The mean value although coincides very well with generallyaccepted values for masonry (Van der Pluijm, 1999: ν=0.20; Lourenço 1996: ν=0.17; Cur 171,1994: ν=0.14-0.28; Hendry, 1998: ν=0.12-0.18).

The mean compressive fracture energy (Gfc = 2.73 Nmm/mm2) is higher than in case of masonryprisms (Gfc = 1.71 Nmm/mm2). From the post peak behavior in Figure 5.10 it can be seen thatthe masonry wallets act less brittle than masonry prisms or cores. Also, the higher peak stresscontributes to a higher fracture energy value as the residual strength remains approximately 1N/mm2. Crack patterns are presented in Annex B.7

119

5.3.4 Influence of test bank and measurement setupBased on the test results from pillars, cores and wallets an estimate of the influence of the testbank stiffness, compression of the ball-and-socket-joint of the Dartec test bank and measurementsetup (LVDT’s) can be made. From the calculated Young’s modulus it is seen that a majordifference is obtained when the values are based on the displacement of the compressive plateon the one hand or the deformations as measured using linear varying deformation transducers(LVDT’s) on the other hand. The difference is caused by several sources. First, values basedon the compressive plate displacement are significantly lower than the values based on the LVDTmeasurements. This is mainly because of the finite stiffness of the test bank and the ball-and-socket-joint in which grease is compressed until steel to steel contact is obtained. On the otherhand, which should not be neglected, the spread on the results based on the LVDT measurementsshow a significantly higher coefficient of variation. When using the test bank displacement, thecoefficient of variation on the results for the Young’s modulus is comparable with the coefficientof variation on the strength results, see Tables 5.3, 5.5, 5.8, 5.10, 5.12, and 5.13. It seems thatusing LVDT’s, an extra measurement uncertainty is introduced. These might be induced by thelateral deformation of the specimens during the compressive test.

Using a linear regression analysis, this uncertainty is quantified. For the analysis, abstraction ismade from the sources that may cause the bias and uncertainty on the results. The regressionanalysis is performed on the derivative quantity that is looked for, namely the Young’s modulusand not on the force-displacement variables. Based on a sample of 26 observations (couplets forYoung’s modulus testing), a linear regression is performed (Y=Ey,LVDT and X=Ey,plate) (Van Dyckand Beirlant, 1999), Figure 5.11. The variance analysis is summarized in Table 5.14.

(5.10)

[ ] [ ]

Y a b Xa b x and

E Y b E X and bY x Y x

Y X

= + × += + × =

= × = +

εµ σ σ

σ σ σε

ε

:

: 2 2 2 2

in which:

(5.11)( )

� ; � � . ; � .

.

a N mm and bSSEn

N mm and SSLSSR

a b= = = =

=−

= = =

390 369 1481 0 42

2408 059

2

2 2 2 2 2

σ σ

σ ρε

Type ofvariance

Variance(Sum of Squares)

degrees offreedom

Mean Variance (MS) F-value Fαα=0.05

modelresidutotal

SSL = (1451)²SSE = (1996)²SSR = (2468)²

1n-2 = 24n-1 = 25

MSL = (459)²MSE = (408)²

1.27 4.26

Table 5.14: Variance analysis - summary

120

0

500

1000

1500

2000

2500

3000

3500

0 200 400 600 800 1000 1200 1400

Ey,plate [N/mm2]

y=1.481x+390ρ=0.59

Pillars

Cores

Wallets

Figure 5.11: Ey,plate versys Ey,LVDT based on test results on pillars, cores and wallets

These results show that there is a bias on the results as the regression coefficients a and b differfrom zero in the mean. Morever, because of the limited amount of results, their value is not thatreliable, certainly not for the intercept a, having a large error (σa = 369 N/mm2). Besides themodel deviation, indeed there is an increase in the spread as suggested, which can be seen fromthe standard deviation on the error term (σε=408 N/mm2).

It is assumed that in the mean, the Young’s modulus measured using LVDT’s, results in thecorrect value. This mean value will be used for further applications. But the extra uncertainty,caused by the measurement setup (σε) is not accounted for as it is not inherent to the materialproperty. Thus, the coefficient of variation based on Young’s moduli calculated from platedisplacement will be taken as representative for the material uncertainty and will be used infurther applications.

5.3.5 Effect of sample size and boundary conditions on the compressive strength fcAs can be seen from former tests, the mean compressive strength differs as a function of the typeof sample used. The mean compressive strength seems to decrease as the “slenderness”increases. The European Standard Eurocode 6 (EC6, 1995) compensates already for thisslenderness on the level of the material properties of the bricks by introducing a normalizingfactor δ. Also in calculating the compressive strength of a wall subjected to a vertical load (NRd),this slenderness should be accounted for (EC6, 1995):

(5.12)Nd f

Rdi m k

m

=Φ , . .γ

121

in which Φi,m is a capacity reduction factor depending on the effect of slenderness and loadeccentricity. The influence of the slenderness is given as the ratio between an effective height(hef) and thickness (d) (EC6, 1995):

(5.13)λ =hdef

The effective height yields:

(5.14)h hef n= ρ in which ρn is a reduction factor that accounts for the stiffening of the wall at the loaded ends.To account for the boundary conditions used in the compressive tests of masonry samples, ρnwith n=2 should be applied as two edges, top and bottom, are stiffened. ρ2 can be one of thefollowing 3 values:

ρ2 = 1 in the common case,ρ2 = 0.75 in case of fully restricted at one edge, (5.15)ρ2 = 0.5 in case of fully restricted at both edges.

Target is to visualize the obtained strength values as a function of a representative slendernessindicator. This is done to estimate the tendency towards decreasing strength values for increasingslenderness. During testing it is seen that not only the smallest dimension of width (w) andthickness (d) has an influence. It is believed that also the largest dimension has an influence onthe results (Ba ant and Cedolin,1991).

For example, a masonry pillar with dimensions w×d×h = 188×188×570 mm is believed to havea lower strength value than a wallet with dimensions w×d×h = 188×577×570 mm. This is notaccounted for in the proposed formula in Eurocode 6, Eq. 5.12. Therefore the formula for theslenderness ratio is adapted in the following sense:

(5.16)λ ρ=×2h

d w

The strength values (fc), geometrical parameters (h, w, d) and boundary condition parameter (ρ2)for the slenderness (λ) of the different types of test samples are summarized in Table 5.15.Besides the experimental data as summarized in the former sections, some data from literatureare used (Schaerlaekens and Schueremans, 1997). These samples are constructed with the samebricks and a similar hybrid mortar composition. The results are visualized in Figure 5.12.

122

0

1

2

3

4

5

6

7

8

1.25 1.45 1.65 1.85 2.05 2.25 2.45 2.65Slenderness: λ=

wallets (n=3)

cores φ=113 mm, h=170 mm (n=6)(*)

pillars (n=19)

cores φ=150mm (n=5)

full scale wall (n=1)(*) : wxhxd=2000x2000x188

Figure 5.12: Compressive strength as a function of slenderness (*: Schaerlaekens andSchueremans, 1997)

summary of own research, section 5.3.1.1-5.3.1.3

literature research (*)

test sample cores‰150h=300mm

pillarsh=360mm

walletsh=570mm

cores ‰113h=300mm

wall 1h=2000mm

[N/mm2] 4.53 4.26 6.07 4.30 3.75xs [N/mm2] 0.77 0.83 0.57 0.59 /

cov [%] 17 19 9.3 13.7 /

n 5 19 3 6 1

PDF / LN / / /

ρ2 1 1 0.75 1 0.75

h [mm] 300 362 570 170 2000

w [mm] 142 188 582 113 2000

d [mm] 142 188 188 113 188

λ 2.11 1.93 1.29 1.50 2.45Table 5.15: Summary of compressive tests on masonry samples - own research and literature results (*: Schaerlaekens and Schueremans, 1997)

123

Assuming that the full scale wall is most representative for masonry as a composite material, thestrength values of pillars and cores are more representative for masonry than small masonrywallets. As pillars and cores are far more easy to build and test, these are preferred. In addition,cores can be taken from an existing building representing the original state of the masonry. Thisis almost impossible for wallets (NBS 62, 1977; Charter of Venice, 1964).

5.4 Compressive strength of masonry - stochastic extension

Different numerical models have been proposed and refined to estimate the compressive strengthof masonry, based on the compressive strength of bricks and mortar (Binda et al., 1988; CUR171, 1994; Cur 92-8, 1982; EC6, 1995; Haseltine, 1986; Hendry, 1998; Kirtshig, 1989; NBNB24-301, 1976; Santos, 1995; SIA V177, 1995; TNO-report BI-78-44, 1978; Van Balen, 1991;Vermeltfoort, 1995). Former research (Claes and Herman, 1999) revealed that the formulaproposed in the European Standard EC6 (EC6, 1995) is one of the most reliable and practicallysuitable numerical relationships. The formula is the result of a statistical regression, based ona large number of test results. In that, it is usable over a wide range of input values, which isparticularly interesting in case of extension to random variables.

This section discusses the effect of using random variables as input variables in theaforementioned numerical model. The experimental research as discussed in former sectionsresulted in the probability distribution functions for brick, mortar and masonry. This enables toverify the numerical results.

The compressive strength (f’k) of masonry according to EC6, based on the compressive strengthof bricks (fb) and mortar (fm) yields:

(5.17)( ) ( )f K f fk b m' . .

.= δ0 65 0 25

The fact that this formula is a product of two factors makes it interesting for stochastic extension.Assumed that fb and fm are lognormally distributed - as determined experimentally - the resultingrandom variable will again be lognormal distributed because of the multiplication model (VanDyck and Beirlant, 1999). Tests on pillars seem to confirm this statement, Section 5.3.1.

In literature, two elements in this formula, Eq. 5.17, are subject of discussion (Santos, 1995):• because of the curve fitting used to estimate the parameters, a dimensional problem

occurs. In case the compressive strength of bricks and mortar are expressed as [N/mm2]and the factor K is a non-dimensional constant, the resulting dimension is no longer thedimension of a stress value [N/mm2]. In Eurocode 6 this problem is solved by assigninga dimension [N/mm0.1] to the constant K. Other authors (Kirtschig, 1989) propose toincrease the exponent of the mortar strength to 0.35. This has the additional advantagethat the (too) limited influence of the mortar strength, certainly in case of lime mortars,is increased a bit (Van Balen, 1991),

• starting from mean values of bricks and mortar a resulting characteristic strength ofmasonry is calculated, which is at least an unfortunate choice. This has its consequencesfor the stochastic extension of the formula and the calculation of mean value and spread.

124

Therefore these authors propose to adapt the formula into following form (Santos, 1995;Kirtschig, 1989):

(5.18)( ) ( )f K f fb m' . .= 0 65 0 35

For further research the original formula is used, despite of the inconveniences as the parametersare optimized for that formula, based on a large number of test samples. This is required whenit will be used for a wide range of variables as in case of a stochastic extension.

For bricks, the normalized compressive strength (EC6, 1995) is used, introducing a normalizingfactor δ. This normalizing factor δ is a function of the brick’s geometry (EC6, 1995; pr ENV772-1,1999). For the test specimens used in this research (cores ‰50 with h = 44 mm andcouplets, 188×88×120 mm), the normalizing factor equals: δ = 0.82 in case of cores ‰50 and δ= 1.11 in case of couplets. Based on the formula proposed in EC6 the characteristic strength ofmasonry can be calculated. The value of the coefficient K for group 1 masonry units equals(EC6, 1995): K = 0.6.

Assuming a lognormal distribution function for the strength of mortar and brick, the numericalmodel can be translated into:

(5.19)( ) ( ) ( ) ( ) ( )ln ln . ln . ln . lnf K f fk b m= + × + × + ×0 65 0 65 0 25δ

Mean value and spread can be calculated (Beirlant and Van Dyck, 1999) using:

(5.20)µ δ µ µ

σ σ σln ln ln

ln ln ln

ln( . ) . ln( ) . .

. .f f f

f f f

k b m

k b m

= + + +

= +

0 60 0 65 0 65 0 25

0 65 0 252 2 2

in which:

(5.21)

( ) ( )( )σ

µσ

µ µ σ

µ µ σ

σ µ σ

ln

ln ln

ln ln

ln

ln

ln

exp

exp

ff

f

f f f

f f f

f f f

2

2

2

2

2 2 2

1

12

12

1

= +�� ��

�

�

��

���

= −

�

�

��

��

⇔= +�

��

��

= −

�

��

�

Using the inverse relations as mentioned on the right hand side of Eq. 5.21, the mean value andspread of the resulting lognormal distribution can be calculated.

As the outcome of the numerical model results in the characteristic strength, Eq. 5.19, the meanvalue can be back calculated using:

125

(5.22)( )µ σ µαα

f k f fk

kmas k mas

f z or fz f

= + × =− ×

:cov( )1

in which zα equals 1.645 for the 95% quantile.

compressivestrength (fc)

PDF µ [N/mm2] σ [N/mm2] cov [%] test samples

brick LN 6.34 2.37 37.4 cores ‰50 (1)

LN 5.16 1.56 30 couplets (2)

mortar LN 8.31 1.79 22 mortar beams

masonry(experimental)

LN 4.26 0.81 19 pillars

/ 4.53 0.77 17 cores ‰150

masonry(numerical)

LN 4.60 1.00 22 numerical - cores ‰50 (1)

LN 4.55 0.84 18 numerical - couplets (2)Table 5.16: Probability distribution type and parameters for masonry compressive strength

It is believed that the formula used in the European Standard, Eq. 5.17, translates theexperimental spread on masonry. Therefore the first option of Eq. 5.22 is used. Using thisequation results in lower mean values and lower spread than the second formula, which lead tohigher mean values and higher spread. The first coincides very well with the experimentalresults. The second formula results in strength values for masonry that are higher than thestrength of the bricks, with a coefficient of variation that is larger than the coefficient of variationon the bricks. This would be very unlikely and is not supported by the experimental results andvalues in literature (Hendry, 1998; EC6, 1995).

The results based on this closed form equations are summarized in Table 5.16. The last two rowscontain the numerical results for the masonry compressive strength. The top row is based on thestrength of cores ‰50 as brick strength input (1), the bottom row is based on the strength ofcouplets as brick strength input (2). Comparing the numerical results with the results based onexperimental research, it is clear that a good estimate of the mean value is obtained: fc = 4.55-4.60 N/mm2 (numerical) against fc =4.26-4.53 N/mm2 (experimental). The numerical values onlyslightly overestimate the experimental values. Further more a very good approximation of thecoefficient of variation is obtained. In both cases, the coefficient of variation is around 20%.Remark that the numerical model confirms the idea of obtaining a more homogeneous compositematerial. The coefficient of variation on the resulting composite masonry is smaller (or equalin a single case) than the coefficient of variation of the components brick and mortar. This couldalready be observed from the experimental results. Using the method of hypothesis testing, nodifference between numerical and experimental value can be proved, nor for the mean value, norfor the spread.

126

5.5 Homogenization - stochastic extension

This section deals with the homogenization technique as proposed by Lourenço (Lourenço, 1996)and adopted in the minutes of the Rilem Committee (Rilem TC MMMN9, 2000). The methodis extended introducing random variables for the material properties and geometry. Target of thisstochastic extension is to enable the estimation of the stiffness of masonry based on the stiffnessof the components brick and mortar.

In Sections 5.2 and 5.3, the probability distribution parameters of the Young’s modulus of brick,mortar and masonry have been estimated. This information will be used to estimate and verifythe Young’s modulus or stiffness matrix of a representative volumetric element (RVE) or basicelement.

Important is to check whether or not this method is able to estimate the average behavior of thecomposite material masonry. The method is outlined in detail elsewhere (Lourenço, 1996) forthe deterministic case and has been subject of extensive research (Schueremans, 1998; Huyse,1999). The question that arises is whether or not the stochastic extension is capable in describingthe probability distribution function of the Young’s modulus of masonry based on the probabilitydistribution function of the components brick and mortar, for a preset three dimensional layoutand adding geometrical uncertainties.

The extension is made for the linear-elastic case. As only the crude Monte Carlo simulationmethod is used, the method can be extended to non-linear material behavior with minor effort(Lourenço, 1996).

5.5.1 Homogenization - State of the artThe technique of homogenization (Bakhvalov and Panasenko, 1989) is becoming increasinglypopular, no longer only for concrete but also for masonry. A method that would permit toestablish constitutive relations in terms of averaged stresses and strains from the geometry andfrom the constitutive relations of the individual components represents a major step forward inmasonry modeling. Given the difficult geometry of the masonry basic cell, a closed formsolution of the homogenization problem is not yet available.

Three different approaches can be distinguished (Rilem TC MMM N9, 2000). The first verypowerful technique is to handle the block structure of masonry by considering the discontinuumwithin the framework of a generalized Cosserat continuum theory (Besdo, 1995; Mühlhaus,1993). The introduction of a characteristic length in the constitutive description allows forlocalization of deformations in a narrow, but finite band of material. In doing so, it is capableof handling the unit-mortar interface and true discontinuum behavior (Lourenço, 1995). Thiselegant and efficient solution possesses some inherent mathematical complexity and has not beenadopted by many researchers. The step towards the practical applicability is still to be given.

A second approach (Anthoine 1995, 1997; Urbanski et al., 1995) is to apply the homogenizationtheory for periodic media to the basic cell. Therefore a single homogenization step is carried outwith adequate boundary conditions and exact geometry. This method has some majordisadvantages. The unit-mortar interface is not accounted for, the complexity of the masonrybasic cell implies a numerical solution of the problem using finite element methods. Therefore,

127

the method is primarily used to determine macro-parameters and not to carry out structuralanalyses.

The third approach can be considered as an “engineering approach” (Lourenço, 1997). The aimis to substitute the complex geometry of the basic cell by a simplified geometry so that a closeform solution of the homogenization problem is possible (Pande et al., 1989; Maier et al., 1991;Pietruszczak and Niu, 1992). The homogenization is performed in two steps, head and bed jointsbeing introduced successively. In this case masonry can be assumed to be a layered material,which simplifies the problem significantly. Lourenço (Lourenço, 1996) further developed theprocedure, presenting a matrix formulation that allows a clear implementation of linear elastichomogenization algorithms, but also a relatively simple extension to nonlinear behavior. Theunit-mortar interface is however not accounted for.

The methods described above result in deterministic constitutive models that provide adescription of the overall or average macroscopic behavior. This is due to the fact thatconstitutive behavior is generally based on a statistical regression between loading and response.Consequently, a straightforward randomization of a given deterministic material model for thepurpose of stochastic structural analysis may not be justified. A more thorough approach is todevelop a consistent framework where the random field characterization of elastic properties isbased on simple micro mechanical models - lattice model or particle model - of the material(Huyse, 1999). Extending the stochastic homogenization techniques for the periodic materialmasonry based on micro-mechanical models is however beyond the scope of this research.Besides, as masonry is a periodic material, this advantage is made use of. In the “engineeringapproach”, the deterministic homogenization model starts from the constitutive relations of unitsand mortar, so on the meso-level, to determine the constitutive relations on macro-scale,introducing random variables. A more thorough approach exists in applying a stochastichomogenization based on a lattice or particle model to the bricks and mortar prior to theengineering approach as proposed by Lourenço. The effects on the overall results (Schueremanset al., 1999a) will be discussed.

5.5.2 Governing relations - 2D elastic formulationThe third approach or “engineering approach” is adopted for stochastic extension. In the presentcontext, the smallest volume element for which a material can be considered macroscopicallyhomogeneous is defined as the representative volume (RVE), Figure 5.13. The size of theinhomogeneities differ for different types of material (Lemaitre and Chaboche, 1985). In caseof concrete, with granulate sizes of 1 cm, the representative volume is considered to be10×10×10 cm (Huyse, 1999; Lemaitre and Chaboche, 1985). In case of masonry, no specificrepresentative volume element has been determined yet. Lourenço adopts different sizes,depending from 10×10×10 cm (Rilem TC MMM N9,2000) to 20×20×20 cm (Lourenço, 1996).Additionally, the term representative volume is frequently mixed with the term basic element,defined as the periodic pattern associated to some frame of reference (Lourenço, 1996). Ofcourse, the latter is not per definition a representative volume. Therefore, in the followinganalysis, the number of bricks involved in a single basic element is taken large enough, so thechosen basic cell (29,6×11,6×18,8 cm) might be more or less a representative volume element(Raffard, 2000).

128

RVE

homogenization

Figure 5.13: Basic element, representative volume element used for homogenization

y

x

1

i

n

...

...L

h i

A

x

ex,i

εx

εx

σx

σx,i

(a) (b) (c)

Figure 5.14: Basic cell. Representative 2D prism for a system of parallel layers

Starting from bricks and joints a homogeneous basic cell is constructed. The two dimensionaltranslation of this process is summarized below. A full description for the three dimensional casecan be found elsewhere (Lourenço, 1996). The process basically consists of two consecutivehomogenization procedures for periodic layered materials (Salamon, 1968; Gerrard, 1982). Thelayered material, Figure 5.14, is built from a periodic system of parallel layers, each of which isassumed to be an isotropic elastic material.

The objective is to obtain a macro or homogenized constitutive relation between homogenizedstresses σh and homogenized strains εh:

(5.23)[ ] [ ]σ ε σ σ σ σ ε ε ε εh h hx y xy x y xywhere and= = =C , : , , , ,

Here Ch is the homogenized stiffness matrix, which will be obtained from the micro-constitutiverelations, the constitutive relations of the bricks and mortar joints. In order to establish the macroconstitutive stress-strain relation, the components of the stress and strain tensors, respectively σjand εj, are defined by the integral over the representative surface as:

129

(5.24)

( )

( )

σ σ σ

ε ε ε

j jA

j iAi

j jA

j iAi

AdA

AdA and

AdA

AdA

i

i

= =

= =

�

�

1 1

1 1

:

where i is the layer number. Auxiliary stresses (tx, ty and txy) and auxiliary strains (ex, ey and exy)are introduced as a deviation measure of the ith layer stress/strain state from the averaged values.Under the given assumptions, the stress and strain components for the ith layer, Figure 5.14, incase of homogenization along the x-axis, read:

(5.25)σ σ σ σ σ σε ε ε ε ε ε

x i x y i y i y i xy i xy i

x i x x i y i y xy i xy xy i

te e

, , , , , ,

, , , , ,

= = + == + = = +

Now let the thickness of the ith layer be hi and the normalized thickness pi be defined as:

(5.26)p hLi

i

i

=

where L is the length of the RVE in the x-direction. Substitution of Eq. 5.26 in Eq. 5.25respectively yields the following conditions for the auxiliary stress and strain components:

(5.27)p t p e p ei y ii

i x ii

i xy ii

, , ,= = =0 0 0

To obtain a matrix formulation of the homogenization theory, the vector of auxiliary stresses tiand the vector of the auxiliary strains ei are defined as:

(5.28)[ ] [ ]t ei x i y i xy i i x i y i xy it t t e e e= =, , , , , ,, , , ,

It is noted that half of the components of the auxiliary stress and strain vectors are zero, see Eq.5.25. The non-zero components are assembled in a vector of unknowns xi, defined as:

(5.29)x P t P ei t i e i= +

where the values of the projection matrix into the stress space Pt and the values of projectionmatrix into the strain space Pe are given in Table 5.17 for the homogenization processes alongx- and y-axis respectively.

Homogenization along x-axis along y-axis

Pt diag[ 0 1 0 ] diag[ 1 0 0 ]

130

Pe diag[ 1 0 1 ] diag[ 0 1 1]Table 5.17: Projection matrices Pt and Pe for homogenization along x- and y-axis

The auxiliary stresses and strains can be redefined as:

(5.30)t P x e P xi t i i e i= =

The stresses and strains in the ith layer read cf. Eq. 5.25:

(5.31)σ σ ε εi i i i= + = +t e

and the linear elastic stress-strain relation, for the ith layer, reads:

(5.32)σ εi i i=C

in which Ci is the stiffness matrix of layer i. Inserting Eq. 5.30 and 5.31 in 5.32, with somealgebraic manipulation yields:

(5.33)( ) ( )x P C P Ci t i e i= − −−1 ε σ

From Eq. 5.31 and Eq. 5.33 yields the relation between averaged stresses and strains. Thehomogenized stiffness matrix Ch reads:

(5.34)( ) ( )C P C P P C P Chi t i e

ii t i e

iip p= −� � −

− −1 1

The basic cell of masonry clearly has a non-layered structure, but different authors (Pande et al.,1989; Papa, 1990; Lourenço, 1996) have suggested an approximate approach based on a two-stephomogenization procedure, under the assumption of layered materials, Figure 5.15.

Due to lack in consistency of the homogenization technique, the resulting moduli will differdepending on the homogenization sequence followed (first homogenization along the x-axis,second homogenization along the y-axis or vice-versa). As the differences are small for a widerange of realistic values of the material properties, the method is considered to be appropriate inmost cases as could be verified experimentally (Lourenço et al., 1998). The sequence leadingto the smallest error - xy homogenization - is used in this work.

131

Homogenizationalong x-axis

Homogenizationalong y-axis

x

y

Figure 5.15: Two-step homogenization procedure

5.5.3 2D - Stochastic extension The structural analysis of historic masonry is affected by substantial uncertainties: physicaluncertainty associated with the stiffness of bricks and mortar joints and geometric uncertaintyregarding the thickness of the head and bed joints (Schueremans 1998; Schueremans et al.,1999a). The way in which the variability is brought into the model may differ, depending on theavailability of detailed information or the required detailing of the model. The method isindependent of the number of random variables that is introduced. The computer time of courseis. For ease of computation, bricks and mortar joints are considered to be isotropic material,although this is not the case in practice as could be seen from tests on bricks, Section 5.2.1.Without loss of generality the method is outlined for the bond used in the experimental researchprogram, namely the Flemish bond.

To account for the physical uncertainty of brick and mortar, their Young’s moduli are translatedinto random variables, Figure 5.16. Each basic cell or RVE contains 9 bricks/units, allconsidered mutually independent, introducing 9 random variables. Only one brick is fullypresent in the RVE (B5), the others are only partly present in the RVE. In between the bricksdifferent mortar joints are distinguished, to account for the physical uncertainty of the mortar ina representative way. Each basic cell or RVE contains 8 mortar joints, 2 bed joints (M3 and M6)and 6 vertical joints (M1, M2, M4, M5, M7 and M8). Similar to the bricks, only a few mortar joints(vertical joints M4, M5) are fully present in one RVE, the other joints are only partly in the RVE.For a bed joint only one random variable is introduced as it is believed that it is realized in onetime and will have strongly correlated properties over the full length of the RVE. This leads toanother 8 mutual independent random variables.

The geometrical uncertainty of the bricks is very limited, Annex B.1. Therefore, all geometricuncertainty is concentrated in the thickness of the joints. Thus, besides the Young’s modulus,also the thickness of the 8 mortar joints is considered random. Both head and bed joints aregiven a mean thickness of 10 mm and a COV of 10%. As the thickness is a non-negative value,a lognormal distribution type is adopted. The parameters for the random variables that areadopted for the analysis, are summarized in Table 5.18. These originate from the experimentalresults as outlined in Section 5.2.

132

B1 B2 B3

B4 B5 B6

B7 B8 B9

M1 M2

M3M4 M5

M6M7 M8

d4 mortar

brick

Figure 5.16: Introducing random variables for geometric and physical uncertainty

Random variables PDF µ cov [%]Brick (prisms) Eb1,...,Eb9 LN 1711 N/mm2 23.0Mortar (mortarbeams)

Em1,...,Em8 N 1398 N/mm2 18.0dm1,...,dm8 LN 10 mm 10

Table 5.18: Input variables and their parameters

A crude Monte Carlo simulation (N = 105) is used to calculate the resulting stiffness matrices,based on the two-step xy-homogenization procedure:

(5.35)( ) ( )µ σC C= ���

���

=�

�

���

����

1623 239 01633 0

676

123 17 0122 0

50symmand

symm

Assuming a plane stress situation, the resulting Young’s modulus can be calculated (ν = 0.19)The results and a comparison with experimental values are summarized in Table 5.19.

The following conclusions can be drawn:• the probability distribution function of the resulting stiffness coefficients Cij of the C-

matrix are very well estimated by a normal distribution type. Because of the number ofadditions in the calculation of the resulting stiffness coefficients of the homogenizedbasic cell, the central limit theorem applies (Van Dyck and Beirlant, 1999),

• the basic element is no longer isotropic although the non-isotropic character remainslimited. This is mainly because the differences between the Young‘s modulus of brickand mortar are small in this particular case. In case these differences increase, theresulting constitutive matrix would reveal a more pronounced anisotropic material,although the constitutive components are isotropic. This could already be concludedfrom the deterministic analysis (Lourenço, 1996),

• the mean value for the Young’s modulus of course is equal to the value calculated in thedeterministic case (Ey = 1596 N/mm2),

• it is obvious that the resulting mean stiffness is somewhere in between the mean stiffnessof brick (Eb = 1717 N/mm2) and mortar (Em = 1398 N/mm2). This no longer leaves a large

133

band. Though the number of experimental values is limited, the Young’s modulus (Ey)tends to be around µ(Ey,experimental)= 1670 N/mm2, which seems not to be supported by thenumerical homogenization procedure (Ey,numerical = 1596 N/mm2). Using hypothesis tests,with a confidence level of 95%, this difference is not significant,

Young’s modulus (Ey) µ [N/mm2] σ [N/mm2] cov [%] # PDF

cores ‰150 1690 372 22 5 /

wallets V1-V3 1642 140 8.5 3 /

full scale wall 1(*) 1587 / / 1 /

pillars 1673 368 22 18 /

xy-homogenization 1596 119 7.5 105 N

(Schaerlaekens en Schueremans 1997: (*) pg. 3/62)Table 5.19: Comparison of Young’s modulus between experimental and numerical results

• important second moments are obtained for the different stiffness moduli. It is strikinghowever that again, the resulting coefficient of variation (cov(Ey) = 7.5 %) is smaller thanthe coefficient of variation on the constitutive components units and bricks. A lessheterogeneous composite material is obtained,

• because use is made of perfectly isotropic constitutive components, the stiffness elementsC13, C23, C31 and C32 are exactly zero. This is not the case when use is made of a micro-mechanical model for the bricks and mortar (Schueremans et al., 1999a). In that case itis observed that indeed the stiffness elements C13, C23, C31 and C32 are only zero in themean, but that they contain a significant spread. This is the prize that is paid for notgoing into detail for the micro-structure of the material components brick and mortar.In finite element modeling however, for example Atena2D (Cervenka Consulting, 2001),a Hookean material is assumed after all. Thus it is not possible to introduce theindividual stiffness coefficients anyway.

As already mentioned, for neighboring basic cells, some of the bricks or mortar joints willoverlap. This introduces a natural spatial correlation between the stiffness moduli of differentbasic cells, from which the auto-correlation function can be retrieved. This is beyond the scopeof this study and is treated in detail elsewhere (Schueremans et al., 1999a).

5.6 Masonry in shear regime

Based on two different test setups, the main masonry material properties for the shear regime aredetermined. Some remarks with regard to different test setups and comparison with literatureresults are made. As for the shear regime, different test setups do exist, Figure 5.17 (CUR 171,1994; Hendry, 1998; Janssen, 1996; Mann, 1989; NBS 62, 1972; prEN 1052-4, 1999; Sahlin,1971; SIA V177, 1995; TNO-78-44, 1978; Toumbakari et al., 1999; Van der Pluijm, 1992;Vermeltfoort and Page, 1981; Vermeltfoort and Raaijmakers, 1992, 1993; Vintzileou and

134

α

σ

σNτ

N

N

TT/2

T/2 α = 45-75°or

or

α

σ0

σNτ

a

b c d e

f g h

F

Figure 5.17: Different test setups for masonry in shear regime (a, b, c and g: TNO-78-44, 1978;d: Van der Pluijm, 1998; f: Vintzileou and Tassios, 1995; h: CUR 171, 1994; Vermeltfoort andRaijmakers, 1992; Raijmakers and Vermeltfoort, 1992)

Tassios, 1995). An overview of test setups is given by Hordyk (Hordyk, 1991) and Van derPluijm (Van der Pluijm, 1990). A “standard racking test” has been published in the ASTM E 72-61. The test setup is similar to Figure 3.1.(f).

For in-site testing, an in situ shear test similar to Figure 3.1.(b) can be performed (Atkinson etal., 1988). The normal or vertical stress, caused by the vertical loading in the structure itself, canbe quantified using the flat-jack methodology (Schaerlaekens and Schueremans, 1997).

Recently, a relatively large research program has been performed including shear tests on walletsin the framework of CUR regulations (CUR 171, 1994; Lourenço, 1996; Van der Pluijm, 1999;Van der Pluijm, 1992; Vermeltfoort and Raijmakers, 1992; Vermeltfoort et al., 1993;Vermeltfoort and Janssen, 1996). As the number of different types of bricks, mortar joints andbonds or layout used in that research program are relatively high, the number of samples withidentical parameters never exceeds 3, the minimum required to have a first impression on thespread of the results. This is true for most research programs. In case of shear tests or tensiletests, Section 5.7, the number of test results is remarkably lower than in case of compressivetests, merely because these tests and their interpretation are far more complex than in case ofcompressive tests. This leads to extra uncertainty on the measurements which is one of thereasons why the spread tends to be higher.

135

5.6.1 Masonry CoresThe first type of test samples that has been used to derive the masonry material properties inshear are cores drilled from wallets. This is done as masonry cores are among the mostconvenient samples in case of existing structures, see Section 5.3.5. The wallets from which thecores are drilled, were constructed similar to and at the same time as the wallets used for sheartests, Section 5.6.2. These cores have the same dimensions as the cores used for the research ofthe compressive regime, namely a diameter of 150 mm and a height of 300 mm. The cores havebeen drilled under an angle of 45b.

From the force equilibrium, see Figure 5.17 (e), (f) for definition of symbols, the relation betweenthe applied compressive stresses (σ), shear stresses (τ) and normal stresses (σn) can bedetermined, following (TNO-78-44, 1978):

(5.36)( ) ( )( )

τ σ α α

σ σ α

= × ×

= ×

sin cos

cosn2

In case α equals 45b, the shear stress (τ) equals the normal stress (σn) on the sliding surface.The shear modulus (G) is calculated using:

(5.37)τ γ=G

the shear angle γ is calculated following:

(5.38)( )( )γ αα

=××

� �arctansin

sin∆hh

in which h is the initial height of the specimen and ∆h the difference in height. The shear modulus (G) is calculated similar to the Young’s modulus between the stress levels:τ = 0.5 MPa and τ = 2/3 τmax , in which τmax is the peak shear stress. The fracture energy (Gf,II)for the mode II (shear) failure (Lourenço, 1996) is calculated as the surface under the stress-sliprelationship, from peak stress and above the residual stress level, also called the friction level(Van der Pluijm, 1999).

The stress-strain relationships for the 5 cores are shown in Figure 5.18. The summary of resultsis given in Table 5.20. Individual test results are given in Annex B.8.

The observed failure modes are very much comparable with the different modes as described byVan der Pluijm (Van der Pluijm, 1999), Figure 5.19: bond failure (a), failure in mortar (b), bondfailure, tensile failure in the bricks (c) and diagonal tensile failure of units (d). Additionally,Figure 5.19 gives a simplified representation of the major crack pattern for the tested cores. Inmost cases a combination of the different failure types did occur. As in non of the cases bondfailure occurs, it is assumed that the bond strength is higher than the brick tensile strength.Another reason for bond failure not to occur is the fact that the bricks contain a groove, addingadditional interlocking resistance to the test samples. This is further elaborated in the next

136

1 2 3 4 5

a bond failure b failure in mortar c bond failure and d diagonal tensile tensile failure in failure of units the bricks

Figure 5.19: Failure mechanisms for shear specimens (Van der Pluijm, 1999) and experimentalresults 1-5

0

0.5

1

1.5

2

2.5

3

3.5

0 0.005 0.01 0.015 0.02 0.02shear angle γ [rad]

Shea

r stre

ss τ

[N/m

m2 ]

200

mm

h =

300

mm

φ = 150 mm

LVDT

Figure 5.18: Stress-strain relationship for masonry cores ‰ = 150 mm, diagonally drilled

section.

137

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.005 0.01 0.015 0.02 0.025shear angle γ [rad]

Shea

r stre

ss τ

[N/m

m2 ]

Wallet D3

Wallet D1

Wallet D2 lref = 300 mm lref = 300 mml re

f = 3

00 m

m

w= 577 m

m

h= 570 mm

w = 300 mm

Figure 5.20: Test setup and stress-strain relationship for masonry wallet shear tests

Masonry cores h×� = 300×150 mm - age 209days - diagonally drilled

τmax[N/mm2]

G [N/mm2]

Gf,II [Nmm/mm2]

sample mean: xsample spread: scoefficient of variation: cov [%]number of test samples: nestimated probability distribution function: PDF

2.430.43185/

883271315/

0.860.57665/

Table 5.20: masonry cores - statistical summary of material parameters

5.6.2 WalletsThree masonry wallets (D1-D3) with nominal sizes: width×depth×height = 600×188×600 mm -Flemish bond - are subject of a shear test similar to the test setup shown in Figure 5.17 (f) and(g). The vertical displacements are recorded using a single linear varying deformationtransducer, with a reference length (lref) equal to 500 mm at front and rear. The horizontaldisplacements are recorded using 2 LVDT’s with a reference length (lref) of 300 mm at front andrear, Figure 5.20. The walls are tested at an age of 158 days. The results of the displacementcontrolled tests (v=0.01mm/s) are summarized in Table 5.21. The τ-γ-diagram is shown in Figure5.20.

The crack patterns of the three wallets do not show fundamental differences. The major crackpattern is schematically illustrated on top of the right hand side corner. The detailed crackpatterns can be seen in Annex B.9. After initial splitting cracks in the bricks and mortar inopposite corners at the edges of the steel compressive blocks, shear cracks occur as well in the

138

mortar joints as in the bricks. In none of the samples loss of bond is observed. The shear stress(τ) and shear angle (γ) are calculated according to Eq. 5.36 and 5.38. Only the area under thecompressive blocks, width equal to 300 mm, is accounted for, as shown in dashed lines in Figure5.20. From the crack pattern in Figure 5.20 it can be observed that this assumption is met quitewell. The two flanges outside the area beneath the compressive block do not influence the stressdistribution, certainly not after the first splitting cracks occur at the corners of the compressiveblocks.

masonry wallets, age 158 days w×t×h = 577×188×570 mm

Wallet D1

Wallet D2

Wallet D3

s covx[%]

τmax [N/mm2] 1.47 1.69 1.33 1.49 0.19 13

G [N/mm2] 623 816 804 748 108 14

Gf,II [Nmm/mm2] 1.52 0.48 1.96 1.32 0.76 58Table 5.21: Masonry wallets shear tests - statistical summary of material parameters

Because of the interlocking resistance after the peak strength, large spread is obtained in the postpeak behavior, resulting in a large spread for the fracture energy (Gf,II). However, the tests onwallets tend to show a less brittle material behavior than in case of cores. The mean value of thefracture energy Gf,II for wallets equals 1.32 Nmm/mm2. For cores Gf,II = 0.86 Nmm/mm2. Similarconclusions could already be drawn from tests in the compressive regime.

The shear modulus very well approximates the numerical value taken from:

(5.39)( )G E=+2 1 ν

In case Ey = 1642 N/mm2 and ν = 0.19, see Table 5.10 for masonry wallets in compression, theshear modulus equals: G = 690 N/mm2. This coincides rather well with the obtained mean value

(G) = 748 N/mm2. Literature values usually report lower values for the fracture energy in casexof mode II (shear) failure (Gf,II) (Lourenço, 1996: Gf,II = 0.05-0.50 Nmm/mm2; Van der Pluijm,1999: Gf,II = 0.06-0.4 Nmm/mm2). Certainly in cases of loss of bond, the fracture energy is low.In this case, the masonry bricks contain a groove. This causes a high friction level in the shearsurface and explains the higher values obtained for the fracture energy on the one hand andspread on the other hand (Page, 1988). Furthermore, the fracture energy in case of mode II(shear) failure (Gf,II) is function of the normal stress (σn) applied. A significant increase is seenfor higher normal stresses applied (Van der Pluijm, 1999).

In several analytical and numerical models, the Coulomb’s friction law is applied to modelmasonry in shear regime (Lourenço 1996; Diana, 1999; Cervenka Consulting, 2001):

(5.40)( )τ σ ϕ= + ×c tan

When literature is reviewed (Van der Pluijm, 1999; CUR 171, 1994; Hendry, 1998), it can be

139

FF/2

F/2

a b c

Failure mechanisms

Different test setups

a b c d

e

f

Figure 5.21: Different tensile tests (a: NEN 3835, 1991; EN 1052-5, 1998; b and c: Vander Pluijm, 1999; e: Copeland and Saxer, 1964; BI-78-44, 1978; f: Schubert, 1976; BI-78-44, 1978) and failure mechanisms (Schubert, 1994)

observed that the friction coefficient for masonry varies within a relatively small range.Moreover, the friction coefficient is more or less independent of the type of brick and mortarchosen: tan(φ) ~LN(0.81,(0.15)2) (Van der Pluijm, 1999). As it can be assumed to be materialindependent, the cohesion coefficient (c) can be back-calculated, using Eq. 5.40 and the testresults (τmax) as summarized in Table 5.21. Based on a Monte Carlo simulation, N = 106 andapplying a log-normal distribution type for the random variables, the resulting cohesioncoefficient equals: c~LN(0.50;(0.15)²). This is an acceptable distribution, lying in an acceptablerange when compared with literature results (CUR, 1994; Van der Pluijm, 1999).

5.7 Masonry in tension

Only recently, the tensile strength of masonry is accounted for (Schueremans and Van Gemert,2000). Masonry has often been considered as a non tension material (NTM) (Heyman, 1966;Smars, 2000; Van Gemert, 1995). Similar to the shear regime, different test setups do exist(Andersen and Held, 1986; BRE 360, 1991; CUR 171, 1994; EN 1052-5, 1998; Hendry, 1998;NEN 6790, 1993; Sahlin, 1971; SIA V177, 1995; Sinha, 1967; TNO-78-44, 1978; Van derPluijm, 1999), Figure 5.21. In the framework of the CUR regulations for masonry based onexperimental research (CUR 171, 1994), a number of tensile tests on masonry are performed(Lourenço, 1996; Van der Pluijm, 1995, 1996 and 1999). But, as the number of different typesof bricks, mortar joints and test setups used in that research program are relatively high, thenumber of samples with identical parameters seldom exceeds 3.

This section will only deal with the direct tensile strength of masonry, needed as input parameterin the finite element modeling, Chapter 6. The flexural tensile strength (ffl) in relation with inplane loading, such as wind loading, is not dealt with in this study. Recent results on flexuralstrength can be found elsewhere (Van der Pluijm, 1996, 1999).

Three different crack mechanisms can be distinguished (Schubert 1994): loss of bond when the

140

tensile bond strength is reached (a), failure through the mortar when the mortar tensile strengthis reached (b) and failure through the brick when the brick tensile strength is reached first (c),Figure 5.21. Of course, in practice and as experienced in the executed experiments, acombination of different failure modes often is obtained, certainly when the tensile strength formortar, brick and the mortar-brick interface are close to each other. As we are only interestedin the tensile strength of the composite masonry, abstraction is made of the type of failuremechanism as they all are a part of the masonry failing.

Different parameters may influence the direct tensile strength and the appearance of one of thepossible fracture mechanisms (Groot, 1987; Hendry, 1998): the composition of the mortar (sand,cement, lime, w/c-ratio, air content, additives), the brick composition (porosity, humidity,roughness of the surface, macro-structure, water absorption), workmanship and test setup. Thus,the extreme variability in the tensile strength test results obtained by most authors is notsurprising.

The 56 test samples according to Figure 5.21 (d), are built at the same time as the masonrywallets. This ensured that the same hybrid mortar composition was used. After sufficienthardening, a circular groove was drilled from on top of the couplets into the brick on the bottom.Using a direct bond test according to prEN-1542 (1998) the tensile strength was determined. Thesamples had an age of 85 days. Table 5.22 summarizes the results. Besides the strength values,the number of occurrence of each type of failure mechanism is indicated too. The individualresults can be consulted in Annex B.10.

Direct tensile strength test samples - � = 50 mm - age 85 days ft [N/mm2]

sample mean: xsample spread: scoefficient of variation: cov [%]number of test samples: nestimated probability distribution function: PDF

0.280.103556trunc. N/LN

failure in the brick/mortar interface: 8; failure in the mortar; 10; failure in the brick: 38Table 5.22: Masonry tensile strength - statistical summary of material parameters

As for the probability distribution type, a normal as well as a lognormal distribution type fitequally well. As the tensile strength values all have the same sign, a lognormal or truncatednormal probability distribution type is preferred. Van der Pluijm proposes to use a normaldistribution type (Van der Pluijm, 1999).

The fracture energy (Gf,I) is determined based on linear regression using literature test results(Van der Pluijm, 1999). The fracture energy is calculated based on the descending branch in thestress (σ)-crack width (wc) diagram (Van der Pluijm, 1999):

(5.41)σf

et

fG

wf

f Ic

=−

,

Based on 129 test samples, the tensile strength versus mode I (tensile) fracture energy for all

141

0

0.0025

0.005

0.0075

0.01

0.0125

0.015

0.0175

0.02

0 0.2 0.4 0.6 0.8Tensile strength ft [N/mm2]

y=0.0148xρ=0.65

Figure 5.22: Tensile strength (ft) versus mode I fracture energy (Gf,I) (Van der Pluijm, 1999)

types of tested samples and the linear regression is shown in Figure 5.22.

Based on this regression curve - forced through the origin to fulfill the definition requirements(Van der Pluijm, 1999) - and the probability distribution for the tensile strength(ft~LN(0.28,(0.10)2) as listed in Table 3.22, an estimate of the fracture energy distribution can becalculated based on the correlation and regression analysis (Van Dyck and Beirlant, 1999: Y =Gf,Iand X = ft): (Gf,I~LN(0.005,(0.0033)2) using:

(5.42)

[ ] [ ]

Y b Xb x and

E Y b E X and bY x Y x

Y X

= × += × =

= × = +

εµ σ σ

σ σ σε

ε

:

: 2 2 2 2

in which:

. (5.43)( )σε2 2

20 003=

−=SSE

n.

The variance analysis is summarized in Table 5.23. Although the F-value is significantly higherthen the critical value Fα, no model fits the data better, because of the wide scatter of the testresults. As tensile test are scarce, no better results are available for the moment.

142

Ground levelFreatic level

σ1=σv

σ3=σc

σ3=σc

AA σ3=σc

A-A(a) (b)

Figure 5.23: Multi-axial stress states in masonry

Type ofvariance

Variance(Sum of Squares)

degrees offreedom

Mean Variance(MS)

F-value Fα(α=0.05)

modelresidutotal

SSL = 0.00083SSE = 0.00117SSR = 0.002

1n-2 = 127n-1 = 128

MSL = 0.000833MSE = 9.17 10-6

90.75 3.92

Table 5.23: Variance analysis - summary

5.8 Masonry in a multi-axial stress state

5.8.1 Introduction

In 1998 the Reyntjens Laboratory acquired a big size triaxial cell testing device for testing lowstrength heterogeneous materials. The goal of the triaxial test setup is to gather informationabout the material behaviour under a multi-axial stress state. In that it simulates the confiningpressure of the surrounding material or external forces. In 1998 and 1999 the test setup wasdeveloped (Swinnen and Schepers, 1999). In 2000 and 2001 test results on the triaxial behaviourof mortar and masonry have been acquired (Duyck and Hallaert, 2001; Hayen et al, 2001a, 2001b;Schueremans et al., 1999b, Lyssens and Vanthilt, 2001).

In the case of masonry, attention is paid to the resulting yield-criterion. This is used in finiteelement models to distinguish the safe from the unsafe area (Lourenço, 1996). Subject of interestis twofold: the capability of this testing device to replace the complex biaxial test setup on smallwallets using brush-plates (Page 1981, 1983) and the (an)-isotropy of the masonry material.

A multi-axial stress state is experienced in several cases, such as: in the crossing of differentthrust lines (a), at foundations due to lateral ground pressure (b), Figure 5.23, or in thick wallsdue to blocked lateral deformations. From other research fields it is known that because of theconfinement, the ultimate multi-axial compressive stress increases. This is experienced for soilmaterials (Van Impe, 1995), chalk or limestone (Homand and Shao, 2000; Bésuelle et al., 2000),rock (Brady and Brown, 1994), concrete (Matthys, 2000), masonry (Page, 1981; 1983; Lourenço,1996) and lime mortar (Raffard, 2000).

143

σ3=σc

σ2=σc

σ1=σv

σ3

σc

σv

σ1=σv

(a) (b)

brick

brick

mortar

Figure 5.24: Triaxial stress state in mortar bed joint

To study the material behavior under confinement, triaxial testing devices are common use in soiland rock mechanics. Although soil and rock are in a triaxial stress state, it is believed that thistype of tests can deliver interesting information concerning the masonry material behavior, evenif masonry is most commonly in a biaxial stress state because of the relatively thin walls.

The test device is also used to look for the mortar behavior on itself. Because of the differentPoissons’ ratio between brick and mortar, the friction in the mortar/brick interface for masonryin compression, Figure 5.24, leads to a certain confinement too. In this case a real triaxial stressstate is developed (Hayen, 1999; Raffard, 2000; Van Hees, 2001). Besides the strengtheningfactor, the increased plastic behavior and deformability are subject of interest. This isdemonstrated, even at relatively low confinement stresses.

5.8.2. Triaxial cell testing device - test setupThe global test setup is visualized in Figure 5.25. The cell is capable of testing cores with adiameter of 150 mm and a height up to 300 mm. Because the main research objective is to studythe behaviour of heterogeneous materials, the size of the samples needs to be rather largecompared to similar test devices used in rock mechanics, Table 5.24. As a consequence, theconfining pressure that can be applied on the material is limited to 17 MPa. The vertical forceis delivered by means of a Dartec universal 5000 kN press. Only compression-compressionstress states can be generated. Thus only information about the multi-axial compression behaviorcan be gathered. The unique part on the triaxial cell is the auto-compensating principle of thehydraulic system, patented by Geodesign (Geodesign, 1997a). The confining chamber isconnected with a self-compensating chamber by means of a small channel in the middle of thepiston. As the confining pressure increases, it applies a force on the bottom of the piston directedupwards. Via the self-compensating chamber, an identical force is applied on top of the piston,downwards, neutralising the upward force. A screw jack pump, called Titan, delivers theconfining pressure. This highly accurate pump can deliver a maximum pressure of 60 MPa. Itsvolume is 300 cm3 with an accuracy of 0.1 mm3 (Geodesign, 1997b). To prevent the oil flowinginto the samples, they are put into a rubber coat.

In the tests that were carried out, a constant ratio between the confining pressure (σ3) and verticalstress (σ1) was applied. This ‘proportional’ load path deviates from the traditional load pathsused in case of soil and rock material, where a constant confinement or deviatoric stress state isapplied. Although, similar load paths are used in case of multi-axial stress states on wall panels(Page 1981, 1983).

144

Mounted triaxial cellRing with radial LVDT’s at 90°Axial LVDT’s at 90°Mortar test sample,rubber coating on topData-acquisitionTitan high pressure pumpINFCP

Figure 5.25: Triaxial cell testing device - test setup

Test samples Triaxial cell with self-compensation chamber

DiameterHeight

d ≅ 150 mm25-300 mm

Range of pistonMax. confining pressureInternal diameter

50 mm17 MPa250 mm

Confining pressure-Titan Data acquisition via Windmill data-acquisition softwareMax. pressureOil volumeOil accuracy

60 MPa300 cm3

0.1 mm3

Vertical LVDT’s (4)Horizontal LVDT’s (4)Internal pressureVertical force

±6.35mm, max. non-lin.: 0.06%±2.54mm, max. non-lin.: 0.21%0-17 Mpa0-5000 kN (Dartec test bank)

Table 5.24: Triaxial cell testing device - technical information

145

To ensure the proportionality between vertical and confinement or lateral stresses, it wasnecessary to link the signal of the confining pressure and the vertical force. Since the Dartec testbank is the most dynamic system, capable in applying a force based on an external signal as well,it was decided to use the internal pressure as master and the vertical force as slave. To obtain thepre-set ratio σ3/σ1, a worksheet was developed in which the parameters of the INFCP (infinityC Process)-regulator/indicator are calculated (Geodesign, 1999). Because the Dartec test bankis used as slave device, the triaxial tests are neither force, neither displacement controlled. Theonly constant between the different triaxial tests is the rotation speed of the Titan pump, thus theoil injection rate used for building up the confining pressure.

5.8.3 Test results - Mortar Cores with diameter 150 mm have been made to study the material behavior of the cement-limemortar (CEMI 42.5R/hydraulic lime/sand/water:158/103/1217/292kg/m3) under confinement.28 cores have been cast in a PVC-tubing. The height has been limited to 120 mm because of theexpected high deformations of the samples. After a period of 28 days of curing in laboratorycircumstances (21°C and 65 %RH), the samples were removed from their PVC-tubing. Toobtain complete carbonation, the samples were subjected to an elevated CO2-content (20% CO2)and temperature (50°C) for a period of 10 days. Table 5.25 lists the stress ratio’s that weremaintained as loading path during testing and the number of samples used for each ratio. Thelast two columns list the number of samples that failed for each ratio and the number of samplesat which the maximum confinement pressure was reached. Only for the higher ratio’s a numberof tests were interrupted before collapse because the maximum confining pressure had beenreached (17 MPa). Previous to the triaxial test, top and bottom of the test samples have beenrectified and the outer surface was coated with parafix in order to prevent oil penetration. Someof the samples were packed in plastic foil. This method was able to restrict the flow of oil intothe pores in an effective manner and improved the removal of the samples after testing. Anadditional rubber coating prevents oil from penetrating into the test sample during testing athigher confining stresses.

Ratio σ3/σ1 number of samples number of samples thatfailed

maximum confiningpressure reached

0.000.050.100.150.250.500.751.00

44444422

44444300

00000122

Table 5.25: Summary of triaxial tests on mortar cores

The failure points are represented in the σ1-σ2=σ3-plane, Figure 5.26. Because of the isotropicmaterial behavior, the values are copied around the first bisector (Lourenço, 1996; Page 1981,1983). Besides the samples that led to failure, also the maximum stresses obtained in the othersamples reaching the maximum allowable confining pressure, are indicated. These values serveas a kind of lower bound. More detailed information can be found in Annex B.11 (Duyck and

146

02468

1012141618202224

0 2 4 6 8 10 12 14 16 18 20 22 24

Legend : ♦ failure no failure

Loading path withconstant σ3/σ1-ratio

σ1(=σV) [MPa]

σ3=σ2 (=σc) [MPa]

Lourenço’s model

Ellipsoide

Fitting in polarcoordinates using leastsquares

0.050.10

0.15

0.25 0.50 0.75 1.00

Figure 5.26: Triaxial yield criterion for hybrid mortar - test results and models

Hallaert, 2001).

From Figure 5.26 it is clear that a significant stress increase is obtained, even at lowconfinements. Different models are used to fit a triaxial yield criterion through these data. Onlythe compression part is of interest. The first model is a Hill type criterion, proposed by Lourenço(Lourenço, 1996):

(5.44)θσ θ σ σ θ σ1 12

2 1 2 3 22 1+ + =

The constants θ1 , θ2 and θ3 are derived from the material parameters, with θ22-4 θ1θ3<0 to ensure

convexity:

(5.45)

θ

θ

θ

11

2 2

245

21

23

2 2 2 2

33

2 2

1 110

0 01

1 1 1 120

110

110

0 0175

1 110

0 01

= = =

= − −� � = − − = −

= = =

°

f

f f f

f

m

m m m

m

,

, , ,

,

.

.

.

Advantage of this model is that the regression parameters can be calculated based on the physicalproperties. The compressive strength according to both axes: fm,1 and fm,2 and the value at 45°

147

(fm,45°) are taken such to obtain an overall acceptable fit. Although they are based on trial anderror, the final representation remains rather poor. Fitting using a least squares algorithm basedon the available test samples does not lead to a convex result. This option is not shown in Figure5.26. The second model is introduced to obtain a better overlap with the experimental results.Therefore an ellipsoid with main axis at the first bisector is used. It is very much comparablewith the former model. The only difference is that the model is based on a more completedescription of an ellipsoid. In that it is an extension of Lourenço’s Hill type criterion.Afterwards it is rotated over 45°:

(5.46)( ) ( )σ θ θ σ σ σ θ θ1 12

2 1 3 3 32

42− + + − =

The regression constants no longer have a direct physical meaning. They are chosen to obtaina close representation of the real behavior:

(5.47)θ θ θ θ1 3 2 45 3 1 7= = = − =. , ,

The final model is based on the experimental results in polar coordinates (r,θ). A least squaresalgorithm is used to obtain an optimal fit in the area of low σ3/σ1-ratio’s, which is important toquantify the strengthening effect at low confining stresses (Duyck and Hallaert, 2001; Hayen etal., 2001a, 2001b):

(5.48)r = − −� �θ θ θ π1 2

2

4

Again, the regression parameters no longer represent a direct physical property of the mortaritself:

(5.49)θ θ1 225 87 3215= =. , .

Using these parameters, an acceptable fit in the area of low σ3/σ1-ratio’s is obtained. This resultsin a workable instrument to determine the strengthening for low confinement. When an averagetensile strength of the brick of ft=0.28 MPa (Table 5.22) is taken as maximum attainableconfinement, Figure 5.24 (b), a multi-axial compressive strength of approximately 6.78 MPa isobtained. Compared with the average uni-axial compressive strength fm,1=5.37 MPa, a strengthincrease of 25% is achieved. When looking at the stress-strain relationships for different σ3/σ1-ratio’s, Figure 5.27, followingobservations can be made:• a change in material behavior from brittle in uni-axial compression to a combined elastic-

plastic behavior in triaxial loading, already at low σ3/σ1-ratio’s (σ3/σ1= 0.05), • an increase in (plastic) deformations with increasing σ3/σ1-ratio’s. When the brick tensile

strength of 0.28 MPa is taken as maximum allowable confining stress, a σ3/σ1-ratio of

148

02468

101214161820

0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1

Vertical strain εV [mm/mm]

σ=

σ3/σ1=0,00

σ3/σ1=0,05

σ3/σ1=0,10

σ3/σ1=0,15

σ3/σ1=0,25

σ3/σ1=0,50

Figure 5.27: Vertical stress-strain relationship for different σ3/σ1-ratio’s

0.043 is reached, when compared with the multi-axial compressive strength of 6.78 MPa.At this ratio, plastic deformations are very plausible what might explain the high plasticdeformations often seen in practice (Smars, 2000; Van Rickstal, 2000; Van Balen et al.,1998),

• a strong increase in ultimate compressive strength. The increase is largest between theuni-axial compression state and σ3/σ1-ratio equal to 0.05. This means that even lowratio’s result in a significant increase in compressive strength.

5.8.4 Test results - masonry For masonry, cores have been drilled from small wallets, Figure 5.28. These were constructedat the same time as the other wallets used for direct testing, Section 5.3.3, to ensure maximumsimilarity of test results.

Main difficulties experienced during testing of the masonry cores were oil infiltrating into thetest samples and reaching the maximum allowable confinement stress (17 MPa) before failure.The masonry cores, drilled from wallets, visually demonstrate several holes, mainly in the mortarjoints but also in the only partially filled groove in the bricks, Figure 5.28. Before applying anexternal rubber coating, these are filled up with a fast hardening gypsum plaster. Although thisprevents the rubber coating from failing at low confining stresses, local failure at more elevatedstresses occurs at initially non visual holes that are just beyond the masonry surface. When theconfining pressure is increased, the thin masonry layer fails. Because of the resulting cavity, therubber coat is pressed into the hole and the fracture strain is exceeded. The rubber coat itselfcracks and oil penetrates into the masonry sample. A sudden drop in oil pressure is caused.Since this is the driving force during the test, a premature test end is caused.

149

Holes in brick groove and mortar jointsFigure 5.28: Masonry test samples after rectification

To avoid this local failure and its premature test ending, different external layers have beentested, Figure 5.29:• to increase the replaceability of broken rubber coating, it is built up from different

partially overlapping strokes of an inner tire, Figure 5.29 (a,b);• the external coating is split up in two parts. One part serves as a layer capable of bridging

eventual local failures. For this purpose, use was made of a thin aluminous sheet (1mm),Figure 5.29 (a) or a thin leaden layer (0.75 mm) which was chosen for its plasticdeformation at low stresses, Figure 5.29 (b). The oil proofing still is provided by a rubbercoating on top. This solution did not prove to be workable in each test. In some tests,the lead itself fissured as well as the rubber on top of it. Nevertheless, the method wascapable in extending the ratio’s σ3/σ1 to higher values;

• final alternative was a fabric made from fibre reinforced PVC, Figure 5.29 (c,d). Thisflexible and pliable textile has a high tensile strength capable of spanning local defectspreventing the external rubber coating from collapse. Although careful workmanship,the fabric did collapse twice during testing, Figure 5.29 (d). Disadvantage of this systemis that, because of the friction between the masonry external surface and the fabric, anadditional confinement is added. The radial displacements are prohibited to a certainextent. In that it has the same effect as an external reinforcement on concrete columns(Matthys, 2000). This effect can hardly be quantified. To prevent this effect fromhappening, it is preferable to use this system only when the lateral displacements arerelatively small. This is in the case of higher σ3/σ1-ratio’s (σ3/σ1A0.75; Duyck andHallaert, 2001), where the confining pressure is more elevated. Fortunately, this is theregion where the extra protective layer is required to prevent local failure and thus toreach global failure.

The external layers have their influence on the measurement of the radial displacements. Theradial displacements are recorded using 4 LVDT’s. These are placed on an external ring, eachat 90° and make a point contact on the external surface of the test sample, Figure 5.25 (a). Theadditional layers that prohibit local failure and provide oil proofing, bias these measurements.Additional bias comes from the fact that the maximum displacements are often not retraced inthe middle of the sample height, but at the bottom or top of the sample. The external ring with

150

Aluminous coatingRubber coating on top

Lead coatingLocal failure

Pliable textileSpanning local failure

Local failureGlobal failure

(a) (b)

(c) (d)

Figure 5.29: Different external layers to prevent local failure and oil penetration

LVDT’s is placed at the middle of the sample height. No strain gauges are glued on the masonrysurface itself. The oil volume change in the triaxial cell is not measured either.

Table 5.26 lists the number of samples that was tested and the number of samples that reachedglobal failure. These are given as function of the stress ratio σ3/σ1. Total number of samplesamounts 30. Between brackets, the number of tests performed is given too. The last twocolumns list the reason in case of premature test ending. When leakage occurred, the test samplewas reused after a sufficiently long period for the oil to seep out. Detailed information can befound in Annex B.12 (Duyck and Hallaert, 2001). The results are presented in the σ1-σ2=σ3-plane, Figure 5.30. The loading paths are indicated in dotted lines. The uni-axial compressivetests, Section 5.3.4 are added on the σ1-axis (σ3=0 MPa).

151

0

2

4

6

8

10

12

14

16

18

20

0 2 4 6 8 10 12 14 16 18 20

Legend : ♦ global failure no global failure

Loading path withconstant σ3/σ1-ratio

σ1(=σV) [MPa]

σ3=σ2 (=σc) [MPa]

Lourenço’s model :

Fitting in polarcoordinates using leastsquares :

0.25 0.40 0.50 0.60 0.75 1.00

2.00

4.00

≅∞

0 018 0 023 0 018 17 53 9

12

1 2 22

1 3 45

. . .. ;, , ,

σ σ σ σ− + == = =°f f MPa f MPam m m

r = − −� �15 50 1814

2

. . θ π

Figure 5.30: Triaxial tests on vertical drilled masonry cores

Ratioσ3/σ1

number of samplestested(tests performed)

number of samplesleading to globalfailure

local failure,leakage

maximumconfining pressurereached

0.250.400.500.600.75124Q

3 (5)3 (3)5 (7)3 (3)6 (10)3 (7)3 (3)2(8)2(3)

313130000

032220211

000013111

Table 5.26: Vertically drilled masonry samples - triaxial testing

From Figure 5.30 it is concluded that:• Because the number of test results is limited, a yield criterion spanning the complete

compression-compression area can not be calculated. The remaining uncertainty is toolarge for that purpose. To be able to estimate the average strength increase for low σ3/σ1-ratio’s, two models are used. The first model is the Hill type criterion according to Eq.5.44, the second is a least squares fit in polar coordinates, Eq. 5.48. The modelparameters are given in Figure 5.30. A least squares algorithm could not be used toretrieve optimal parameters for the Hill type criterion. The required values at higherσ3/σ1-ratio’s are missing for that purpose.

• Although the limited number of successful test results, the strengthening, even at low

152

02468101214161820

0 2 4 6 8 10 12 14 16 18 20

Legend : ♦ global failure no global failure

Loading path withconstant σ3/σ1-ratio

σ1(=σV) [MPa]

σ3=σ2 (=σc) [MPa]

Lourenço’s model:

Fitting in polar coordinatesusing least squares :

0.25 0.50 0.75 1.00

4.00

≅∞

0 044 0 078 0 044 14 75 9 8

12

1 3 32

1 3 45

. . .. ; ., , ,

σ σ σ σ− + == = =f f MPa f MPam m m

r = − −20 4 24 94

2. . ( )θ π

Figure 5.31: Triaxial tests on diagonal drilled masonry cores

confining pressure is significant. When a confining pressure of 2 MPa is present, theaverage compressive strength is estimated to be double.

• Also in case of masonry cores, an increased plastic behavior is observed, similar to Figure5.27. When a ratio σ3/σ1=0.4 is reached, a plastic phase is developed after the elasticdeformation (Duyck and Hallaert, 2001).

To study the effect of shear failure in combination with a confining pressure and to have an ideaof the effect of the orientation of the bed joints (Brady and Brown, 1994; Page 1981, 1983), alimited number of masonry cores was drilled under an angle of 45°, Figure 5.28. Table 5.27 liststhe number of samples that was tested and the number of samples that reached global failure.These are given as function of the stress ratio σ3/σ1. Between brackets, the number of testsperformed is given too. The last two columns list the reason in case of premature test ending.For individual test results, the reader is referred to Annex B.13.

Ratioσ3/σ1

number of samplestested(tests performed)

number of samplesleading to globalfailure

local failure,leakage

maximumconfining pressurereached

0.250.500.7514Q

3 (3)3 (4)3 (6)3 (6)1 (1)2 (2)

331000

000002

002310

Table 5.27: Masonry samples diagonal - triaxial testing

153

-10123456789

101112

-1 0 1 2 3 4 5 6 7 8 9 10 11 12-10123456789

101112

-1 0 1 2 3 4 5 6 7 8 9 10 11 12

σ1

σ2 σ2

σ1

σ1σ2

σ1

σ2

[ ]0 013 0 017 0 016 1

8 74 8 03 9 2012

1 2 22

1 2 45

. . .. ; . ; ., , ,

σ σ σ σ− + == = =°f f f MPam m m [ ]

0 036 0 059 0 036 15 3 9 09

12

1 2 22

1 2 45

. . .. ; ., , ,

σ σ σ σ− + == = =°f f f MPam m m

Figure 5.32: Comparison between Lourenço’s Hill Type criterion and Page’s experimentalresults on small masonry panels [adopted from (Lourenço, 1996) and (Page 1981, 1983)]

The test results are shown in the σ1-σ2=σ3-plane, Figure 5.31. The uni-axial test results, Section5.3.4, are added on the σ1-axes (σ3=0). Again both a Hill type criterion, Eq. 5.44, and a leastsquares fit in polar coordinates, Eq. 5.48, are given to estimate the strengthening for low σ3/σ1-ratio’s, Figure 5.31.

From Figure 5.31 following conclusions are drawn:• Although the limited number of successful test results, the strengthening at low confining

pressures is significant and similar to vertically drilled cores.• At low confining pressures (σ3/σ1 @0.5), the fracture pattern is comparable to uni-axial

shear tests, see Figure 5.29(c) and Figure 5.18. When the σ3/σ1-ratio increases (0.75-1.0),the crack direction tend to rotate towards horizontal cracks (Duyck and Hallaert, 2001).

When the obtained test results are compared with the biaxial tests on small masonry panels(Page, 1981, 1983), Figure 5.32, following remarks can be made:• The experimental results do not allow to exclude anisotropic behavior. Page’s results on

plane masonry with full bricks reveals a certain anisotropic behavior. Although thematerial behavior is not perfectly isotropic, Lourenço’s model, based on least squaresestimate, is quasi-isotropic. The ratio fm,1/fm,2=1.09. Thus, the error made when assumingisotropic behavior, remains limited. In case of hollow blocks, the ratio can besignificantly higher, even up to 4.06 (Lourenço, 1996).

• The strengthening at low ratio’s is not covered very well using Lourenço’s model, seealso Figure 5.32 left hand side for σ2K0, because it is a least square fitting for the fullcompression area.

154

5.9 Conclusions

This chapter gives an overview of the experimental research program that was performed toderive the material properties distribution functions. These will be used for the material inputvariables in the reliability analysis. They are required to feed the material models such as a finiteelement analysis. The random variables and their parameters are summarized in Table 5.28.Besides the elastic properties, distinction is made between masonry in compressive, shear andtensile regime.

material property Unit Mean µ Stdev σ PDF

elastic E [N/mm2] 1600 120 N

ν [Nmm/mm2] 0.19 0.06 LN

G [N/mm2] 748 108 LN

compression fc [N/mm2] 4.50 0.85 LN

Gfc [Nmm/mm2] 2.73 0.60 LN

shear c [N/mm2] 0.50 0.15 LN

tan(φ) / 0.81 0.15 LN

Gf,II [Nmm/mm2] 1.32 0.76 LN

tension ft [N/mm2] 0.28 0.10 LN

Gf,I [Nmm/mm2] 0.05 0.03 LNTable 5.28: Masonry material properties - distribution function and parameters. Composition:brick: Kempenbrand module 50, mortar: cement-lime mortar (Section 5.2.6), bond: Flemishbond.

The correlation structure based on the experimental and literature results, equals (only non-zeroelements are filled out):

(5.50)

( )

( )

ρ

ν ϕ

ν

ϕ

=

������������

������������

E f G c G f GE

fGc

Gf

G

s y m m

c fc f II t f I

c

fc

f II

t

f I

tan

tan

. .

.

..

, ,

,

,

1 0 72 0 491

1 0 661

11

11 0 65

1

155

However, it must be stressed again that obtaining representative values is hard to achieve inpractice. The influence of the test setup and measurement method are not negligible. Moreoverthe error made can not always be quantified very precisely. Besides, it is wishful thinking thatfor each safety assessment of an historical building the amount of test results will be availableas it is the case in this study. Therefore, in the future, databases should be made more reliableand become a valid alternative for test results. This will gain ground in the future. Numericalmodels tend to be more representative for the real material behavior, but only if the extraexperimental information needed as input is available. Often more parameters are required,which puts an increased emphasis on experimental research.

Extra complexity and uncertainty in case of existing buildings come from the fact that it willnever be possible to determine the material properties from the building itself to their full extent.In many cases, information on the workmanship and in site conditions can hardly be quantified.

When the different samples are compared, it is concluded that drilled cores are most suitable fortesting purposes. The degree of damage remains acceptable, the representability of the resultsis comparable to full scale walls. Small wallets seem to overestimate the real strength.

In case the degree of destruction by coring is not acceptable, it has been demonstrated thatnumerical models for compressive strength (EC6, 1995) and stiffness of masonry, usinghomogenization techniques, are a valid alternative. But they also need information on brick andmortar properties.

Triaxial testing did prove to be a tedious job. On average, one single test per day is performed,which does not make them practically suitable to gather numerous data required for statisticalprocessing. Practical difficulties prohibit serial testing and affect the reliability of the obtainedresults. Due to leakage in case of heterogeneous masonry cores and reaching the maximumconfining pressure in case of mortar and masonry cores, it was not possible to derive reliable testresults for high σ3/σ1-ratio’s. As a result, the anisotropic behavior of masonry can not be judgedfrom an experimental point of view at this stage. Biaxial test results in literature on smallmasonry panels on comparable full brick masonry samples do show a limited degree ofanisotropic behavior. Using a Hill type yield criterion based on a least square fitting shows a10% difference for the mono-axial compressive strength values according to both principal axes.Compressive tests on the bricks showed significant anisotropic behavior. The homogenizationprocess also reveals a limited anisotropic behavior of the Young’s modulus.

For low σ3/σ1-ratio’s, a significant strength increase is demonstrated both for mortar and masonrysamples, which is quantified using a least squares fit. Furthermore, both the hybrid mortar andmasonry samples show a change in material behavior from quasi-brittle to elastic-plastic withincreasing σ3/σ1-ratio’s. These large plastic deformations already take place at low σ3/σ1-ratio’s.It is believed that this multi-axial stress state can be developed within the mortar joint with anaverage brick tensile strength as low as: ft = 0.28 N/mm², Figure 5.21.

156

157

0.09 m 0.09 m0.22 m

Vext

Vinf

V0=30%

Figure 6.1: Three-leaf stone masonry walls [adopted from (Toumbakari et al., 1999)]

6 Applications

6.1. Introduction

This chapter contains several examples in which the outlined methods are illustrated. Use ismade of the different reliability methods, outlined in Chapter 2 and Chapter 3 and the structuralmodels for masonry, Chapter 4. The material parameters are mainly based on the values obtainedfrom the experimental research, Chapter 5. As the different reliability methods can handleanalytical problems with ease, this is illustrated for the case of grout injection, Application 1, anda masonry column, subjected to an eccentric applied vertical load, Application 2. Of course,these are not the only problems dealt with in practice. Often, problems are much morecomplicated and can no longer be translated into analytical relations, such as in the case ofmasonry arches, Application 3. For these problems, use is made of an external program, todistinguish the safe from the unsafe region. For continuous problems, use is made of the finiteelement code Calfem and Atena2D. This is illustrated for a masonry shear wall, Application 4.To obtain a stochastic finite element method, the reliability analysis is encapsulated in the finiteelement code Calfem, Annex A.15.

6.2. Consolidation of three-leaf masonry walls - grout injection

6.2.1. Problem definitionThree-leaf stone or brick masonry walls are common use in historical buildings (Binda et al.,1994; De Raeymaecker, 2001), Figure 6.1. Often the load bearing capacity of this structuralelement in compression is considered insufficient. This is mainly due to (Valluzzi et al., 2001):• the use of poor materials (low strength lime mortars, presence of voids, mainly

concentrated in a loose internal core),• the irregular geometry,• the lack of connection between the different leafs.

158

In these cases, consolidation using a grout injection may offer an outcome. Extensive researchhas been conducted in this area (Toumbakari et al., 2000; Valluzzi, 2000; Van Rickstal, 2000;Venderickx, 1999), not only to study the strength increase due to the injection of a new material(Tassios, 1995; Valluzzi, 2000), but also to study the optimal parameters (Van Rickstal, 2000),material compatibility (Toumbakari et al., 2000) and quality control using non-destructive testmethods (Venderickx, 1999).

6.2.2. Material modelThe original (unstrengthened) compressive strength of the masonry wall can be estimatedaccording to (Egermann, 1993):

(6.1)f VV

fVV

fwcext

ext,inf

inf,0 0= � � + ���

��

where:fwc,0: the compressive strength of the three-leaf wall, assumed lognormal distributed

fwc,0:~LN(µ(fwc,0),σ(fwc,0)2), fext: the compressive strength of the external leafs, which is assumed lognormal distributed

fext:~LN(µ(fext),σ(fext)2), finf,0: the compressive strength of the internal layer, built up with loose rubble material,

again assumed lognormal distributed finf,0:~LN(µ(finf,0),σ(finf,0)2),Vext: the volume of the external leafs,Vinf: the volume of the internal leaf,V: the total volume of the wall.

According to Egermann’s results, Vintzileou and Tassios (Vintzileou and Tassios, 1995) assumethat, for the original wall, the compressive strength is mainly due to the external layers, so theinternal core contribution is negligible. On the contrary, the strength increase after injection ismainly due to the infill consolidation, so the contribution of the external layers can be neglected.

In general following relation applies (Vintzileou and Tassios, 1995):

(6.2)f VV

fVV

fwc sext

ext s,inf

inf,= � � + ���

��

where:fwc,s: the compressive strength of the consolidated three-leaf wall,finf,s: the compressive strength of the consolidated infill material.

The compressive strength of the infill material, after grout injection (finf,inj) is determinedexperimentally, based on three-leaf stone/brick masonry walls, injected by cement or multi blendgrouts (Toumbakari et al., 2000; Valluzzi, 2000). Different formula’s have been proposed.These are all based on the compressive strength of the grout (fgr). The formula proposed byTassios and Vintzileou (Tassios and Vintzileou, 1995), reads:

159

(6.3)f finj grinf,..=125 0 5

The formula, presented by Valluzzi (Valluzzi, 2000), reads:

(6.4)f finj grinf,..= 0 31 1 18

Eq. 6.3 and 6.4 are obtained by regression analysis. It is noticed that these formula’s can be usedto predict the ultimate compressive strength, making an error smaller than 20-25% (Valluzzi,2000). To account for this error, a normal distributed model uncertainty (ε) is introduced(Ditlevsen, 1982):

(6.5)f VV

fVV

fwc sext

ext s,inf

inf,= � � + ���

��

�

��

��ε

As there is no systematic bias observed, the mean value (µ(ε)) equals 1. The standard deviation(σ(ε)) is taken 0.16. In doing so, an error of 25% is assumed to encapsulate the outcome withina 95% confidence level.

This approach is very interesting in practice because the main parameters can be evaluated bysimple in situ and laboratory tests. The in situ strength of the external leafs can be estimatedusing the double flat jack method (Binda, 1999; Schaerlaekens and Schueremans, 1997; Valuzzi,2000). A survey of the geometrical characteristics of the section of the wall (visual inspection,cores, endoscopy) allow to define the volumetric parameters. Finally, compressive tests on theinjection grout provide the reference strength for the consolidation material.

In practice, the increase of the average strength of the masonry or the reduction of the varianceof the strength, or both, are the intention of a consolidation injection. These are the parametersof the compressive strength that affect the reliability of the masonry to a certain extent. As theincrease of strength is related to the strength increase of the infill material, this is furtherelaborated.

In practice, the final result will depend on the quality of the injection. This is mainly governedby the degree of filling (Van Gemert and Schueremans, 1997; Van Rickstall, 1999). It issupposed that the average strength of the consolidated infill (finf,s) depends upon the averagestrength of the original infill (finf,0) material and the average strength of the injected infill (finf,inj).Both materials contribute to the global strength in relation with the injected volume (Claes andHermans, 1999; Van Gemert and Schueremans, 1997):

(6.6)fV V

Vf

VV

fsinj

oinj

injinf, inf, inf,=−

+0

0 0

where:finf,s: the strength of the infill material after injection, which is on its turn a random

160

variable finf,s:~LN(µ(finf,s),σ(finf,s)2),finf,o: the original strength of the infill material, also a random variable

finf,o:~LN(µ(finf,o),σ(finf,o)2),finf,inj: the strength of the injected infill material, again a random variable,

finf,inj:~LN(µ(finf,inj),σ(finf,inj)2),V0: volume of voids in the masonry,Vinj: injected volume.

It is obvious from the above expression that:• Seen the negligible strength of the original infill material (finf,0), the higher the injected

volume, the more the average strength of the injected material will increase for constantaverage compressive strength of the original infill material and of the injection grout.This emphasizes the importance of a uniform filling of all voids.

• The higher the mechanical strength of the grout and thus of the injected infill material,the higher the resulting average strength of the infill material on the condition that theinjected volume remains te same.

Experimental research (Toumbakari, 2000; Van Gemert et al., 1998; Van Rickstal, 2000) pointsout that grout injection also reduces the variance of the resulting strength as it makes the resultingmaterial more homogeneous. This is due to a uniform filling of the masonry by the grout on theone hand and a better internal cohesion of the masonry on the other hand. The final variance thusdepends not only on the degree of filling, but should also depend on the variance of the strengthof the grout. These reflections can be retraced from Eq. 6.6:• the better the injection is performed, the higher the injected volume (Vinj). This increases

the relative importance of the variance of the strength of the injected material (Vinf,inj). Ifthis variance is smaller, which is likely in practice, the effect is enhanced.

6.2.3. Reliability analysisImagine a three-leaf brick masonry wall subjected to a vertical permanent load (S~N(1,(0.07)2),normally distributed (Melchers, 1999). Then the limit state function can be outlined, following:

(6.7)( )g R S R S, = −

For the strength value (R), use is made of Eq. 6.5. For the compressive strength of the externalleafs, the experimental values as reported in Chapter 5 are used (fext~LN(4.26,0.812). The randomvariables and their parameters are listed in Table 6.1. For the grout, the experimental results ofa multi-blend mix are used (Claes and Hermans, 1999; Van Gemert et al. 1999).

161

Random variables parameter meanvalueµ

standarddeviationσ

cov [%] probabilitydensity functionPDF

vertical loadexternal leaf(1)

internal leaf(2)

injection grout(2)

voids in infillinjected volumemodel uncertainty(3)

S [N/mm2]fext [N/mm2]finf [N/mm2]fgr [N/mm2]V0 [%]Vinj [%]ε

1.04.260.5112.0530301

0.070.810.401.04//0.15

719789//15

NLNLNLNfixedfixedN

(1): Experimental results, Chapter 5; (2): Toumbakari et al., 1995; Claes and Hermans, 1999;(3): Valluzzi et al., 2001

Table 6.1: Variables and their parameters

Table 6.2 summarizes the results for the reliability analysis on the original problem. As the limitstate function is known analytically, FORM/SORM and ISMC are used to calculate thereliability index. For the consolidated walls, use is made of Eq. 6.3 and 6.4. It is assumed thatthe filling of the voids is perfect. Some experimental values are added for illustration, to indicatethe original strength of the three-leaf masonry wall and its strength after consolidation. Althoughthe strength increase is limited, the reliability index increases more significantly. Besides theaverage strength increase, the effect of the variance reduction affects the reliability index.

Reliability method

β1(1) (pf1) β2(2)(pf2) β3

(3)(pf3) #LSFE

FORMSORMISMC

2.81 (2.3 10-3)2.84 (2.2 10-2)2.85

5.03 (2.5 10-7)5.035.03

5.29 (6.7 10-8)5.295.28

34-4958-7010000

Compressivestrength

Before consolidation After consolidation

fc [MPa](4) 2.092.41

3.152.91

(1): before consolidation; (2): according to Eq.: 6.3 (Vintzileou and Tassios, 1995); (3):according to Eq 6.4. (Valluzzi, 2001); (4): (Van Gemert et al., 1999), Remark: the strength afterconsolidation is biased to a certain extent because the walls have been subjected to a compressive test before

Table 6.2: Random variables and outcome for different reliability methods

Of course, the filling of voids with grout might not be perfect in practice. Figure 6.2 summarizesthe evolution of the reliability index as function of the degree of filling and the materialparameters of the injection product. The results are based on Eq. 6.4. The initial volume ofvoids is taken V0=30%, Figure 6.2, as it was determined experimentally (Claes and Hermans,1999) for this type of filling material. Of course, in practice, the analysis will be carried out forthe actual volume of voids as determined on experimental basis. From the resulting figure, it is

162

Reliability of grout injection, initial volume of voids V0 = 30%

33,23,43,63,8

44,24,44,64,8

5

2 4 6 8 10 12 14 16Average compressive strength of injection material: µ(fgr)

β

V(fgr)=0,10V(fgr)=0,15V(fgr)=0,20V(fgr)=0,25

coefficient of variation of injection material

βT=3,7

Vinj=20%Vinj=25%

Vinj=30%

Figure 6.2: Consolidation of masonry using grout-injection

possible to check the degree of filling and the properties of grout that are needed to reach a presettarget reliability index, e.g. βT = 3.7 (EC1, 1994), for a given initial volume of voids (V0). Thesegraphs can be used as an evaluation after the consolidation has been done.

From Figure 6.2 it is clear that the average strength of the injection material is not the mainissue. This is followed in the design of injection products. Initially, high strength epoxy groutswere used (µ(fgr)=30-40 N/mm2). These products were chosen as they were able to penetrate intosmall voids without problem, because of their low viscosity. The rheology and stability ofcementitious products was of minor quality and was not capable of filling small voids. Also forreasons of cost, the emphasis is shifted towards the rheology, stability and chemical compatibilityof the grouts. Using fine granulates, bentonite, ultra-fine admixtures, additives and high speedmixing procedures, a product is obtained which has similar or even better rheological propertiesthan epoxy grouts. The extra benefit is that these products have a similar strength valuecompared to the original masonry and are chemically compatible with the original material.Thus, a smaller resulting spread in strength values is obtained (Van Rickstal, 2000).

The influence of the coefficient of variation of the injection material on the resulting reliabilityindex is rather small. This is mainly because a coefficient of variation (V(fgr)) of 0.1 up to 0.25is always small in relation to 0.78, the coefficient of variation of the initial filling material, seeTable 6.1 and Eq. 6.3, 6.4 in combination with Eq. 6.6. On the other hand, the model uncertaintyhas a significant influence on the results, see Eq. 6.5. This outshines the gain of the quality ofthe injection grout. In general, following rule applies: taken an average strength value for theinjection product, a smaller spread on the strength will result in a higher reliability index. Thus,a good quality control in practice, leading to low spread on the material properties, should beaimed at.

163

Graphs such as Figure 6.2 can be used for quality control. As the injection has been performed,it can be checked whether or not the preset target reliability is reached. Based on the originalvolume of voids (V0), the injected volume (Vinj) and material parameters of the injected grout(µ(fgr),V(fgr)), the obtained reliability (β) can be derived.

In ideal circumstances, where the total volume of voids (V0=30%) is filled (Vi=30%), reachinga high quality injection product on site (V(fgr)=0.1), an average compressive strength of the groutµ(fgr)=3.25 N/mm2 is sufficient.

6.2.4. ConclusionsThe benefit on the overall reliability of a grout injection has been evaluated using a level IIImethod. It has been demonstrated that not only the mean value, but also the spread on thecompressive strength of the grout has an influence on the resulting strength and thus safety.

For the resulting strength of the injected wall, expressions from literature are taken. Theserelations are based on a limited number of experimental results. A model uncertainty thereforeis included. Of course, whenever a more accurate relation is available, an update can be made.The new model and a smaller model uncertainty can be implemented. The emphasis was noton the material model used, but on the possibilities of the methodology using an available model.

Because the limit state function is a simple expression, which is available analytically, traditionalreliability methods such as FORM/SORM are preferred above sampling procedures such as MC,DS or derivatives. These are much more efficient in this case. The number of direct limit statefunction evaluations remains small, Table 6.2.

If the initial volume of voids is available, design graphs can be established. In practice, these canbe used to adopt the injection parameters in order to meet the safety or reliability (βT) as presetin standards.

6.3. Masonry column subjected to an eccentric vertical load

6.3.1. Problem definitionA masonry column with rectangular cross-section is subjected to an eccentric vertical load,Figure 6.3. The random variables that govern the problem are given in Table 6.3. The materialproperties correspond with the experimental results, Chapter 6. The structural model is describedin detail above, see Chapter 4.

164

d

FV

e

wFrontview

Top view

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Eccentricity e [mm]

e=d/6Com

pres

sive

stre

ss σ

[N/m

m2 ]

FVe

fc

dc dpl

ElasticplasticmodelFVe

ft

dc

fcElasticbrittlemodel

NTM

ft≠0

Figure 6.3: Problem definition - eccentrically loaded column

Random variables parameter meanvalueµ

standarddeviationσ

cov [%] probabilitydensity functionPDF

vertical loadeccentricitywidththicknesscompressive strengthtensile strength

FV [N]e [mm]w [mm]d [mm]fc [N/mm2]ft [N/mm2]

1400001506006004.260.28

140002530300.810.11

1017551939

NNNNLNLN

Table 6.3: Random variables and their parameters

The ultimate limit state that will be checked is the limiting compressive strength of the masonry.Two models will be used, according to the material behavior, explained in Section 4.2. The firstmodel will not account on the plastic behavior of the masonry. This can be looked at as themodel proposed by the code NBN B24-301 (1980). The second model will account on theelastic-plastic behavior of the masonry, according to EC6 (1995). The extra gain in safety usingthis second model, will be discussed. Thus, the limit state function reads:

(6.8)( ) ( )g F e w d f f f F e w d fULS V c t c c V t, , , , , , , , ,= − σ

Figure 6.4 represents the evolution of the limit state function as a function of the eccentricity.The mean value is taken for the other parameters involved.

165

-10

-8

-6

-4

-2

0

2

4

6

0 50 100 150 200 250 300 350

eccentricity e [mm]

g fULS c c= −σ

g fSLS t t= −σSafe

Unsafe

NTM

Limited tensile strength

Elastic-plastic

Elastic-brittle

Figure 6.4: Ultimate and Serviceability Limit State function versus eccentricity e

The compressive stress σc depends on the structural model used. The behavior of this limit statefunction is visualized in Figure 6.4.

The serviceability limit state that will be checked is the occurrence of cracks. First, use will bemade of a pure NTM-material, secondly, the limited tensile strength of the masonry will beaccounted on. The serviceability limit state reads, Figure 6.4:

(6.9)( )g f F e w d fSLS t t V t= − σ , , , ,

Parallel with the structural items, some numerical details are commented, such as the efficiencyof the reliability method used and the sensitivity towards the random variables that govern thelimit state function.

The parameters in Table 6.3 are chosen such that the reliability index approximates the targetreliability index βT= 3.7 for the ultimate limit state, not accounting on the plastic behavior of themasonry.

6.3.2. Ultimate Limit State - probabilistic evaluation on level IIIResults for the elastic-brittle model are summarized in Table 6.4. The results for the secondstructural model are summarized in Table 6.5. Different reliability methods are used to comparetheir efficiency. Some numerical details on the DARS and MCARS+VI procedure are given inTable 6.6.

166

Method β pf YLSFE accuracy

exact valueMVFMFORM NLPQLFORM RFLSSORMMC (Adsamp)VIMC (σh =3)DSDARS (λadd=var)MCARS+VI (λadd=3.00)MCARS+VI (∆g,add=var)

3.662.573.693.693.663.663.693.653.663.663.64

1.26 10-4

5.08 10-3

1.12 10-4

1.12 10-4

1.26 10-4

1.26 10-4

1.11 10-4

1.31 10-4

1.24 10-4

1.24 10-4

1.36 10-4

/7445064/70100005000014432482332

/////V(pf)=2.5% V(pf)=31%V(pf)=47%,σ(β)=0.17,V(β)=0.04V(pf)=51%, σ(β)=0.17,V(β)=0.04V(pf)=51%,=σ(β)=0.17,V(β)=0.04V(pf)=42%,=σ(β)=0.17,V(β)=0.04

Table 6.4: Summary of results - reliability analysis - elastic-brittle model

Method β pf YLSFE

accuracy

exact valueMVFMFORM NLPQLFORM RFLSSORMMC (Adsamp)VIMC (σh =3)DSDARS (λadd=var)MCARS+VI (λadd=3.00)MCARS+VI (∆g,add=var)

4.062.784.084.084.064.064.064.094.164.064.08

2.45 10-5

2.72 10-3

2.25 10-5

2.25 10-5

2.45 10-5

2.45 10-5

2.46 10-5

2.13 10-5

1.60 10-5

2.44 10-5

2.25 10-5

/7445064/70100005000021381112923

/////V(pf)= 2.5%V(pf)= 18.7%V(pf)= 163%,σ(β)=0.13,V(β)=0.03V(pf)= 67%,σ(β)=0.09 ,V(β)=0.02V(pf)= 67%,σ(β)=0.09 ,V(β)=0.02V(pf)= 67%,σ(β)=0.09 ,V(β)=0.02

Table 6.5: Summary of results - reliability analysis - elastic-plastic model

The difference between the two models results in almost an order of magnitude difference infailure probability. For the ultimate limit state, accounting on the tensile strength of the masonrydoes not lead to a significant increase in reliability index. This already can be seen from Figure6.3. In that, the two graphs almost overlap near failure.

The different reliability procedures find an accurate estimate of the reliability index.MCARS+VI is most efficient in this example. The difference between MCARS+VI (λadd=3.00)and MCARS+VI (∆g,add=var) is limited. The final additional distance on the outcome space, islisted in Table 6.6. In the same row, the final additional distance λadd for the DARS (λadd=var)is mentioned. It is seen that this final value is significantly smaller than the value preset byWaarts, λadd=3.00. In all cases, a pure quadratic response surface was chosen to fulfill theminimum error requirement.

167

Reliabilitymethod

elastic model elastic-plastic model

DARS

λadd=var

MCARS+VIλadd=3.00

MCARS+VI∆g,add=var

DARS

λadd=var

MCARS+VIλadd=3.00

MCARS+VI∆g,add=var

λADI,min

λmin

βRS,SORM

λadd,fin//∆g,add

σh,finRS-modelF/Fα

4.513.783.771.28/p-qK0

4.51/3.663.001.79p-qK0

4.51/3.721.981.78p-qK10-17

4.864.464.170.88/p-qK0

4.86/3.293.001.98p-qK0

4.86/3.580.401.99p-qK10-23

legend: RS-model: p-q for pure quadraticTable 6.6: Numerical details DARS and MCARS+VI

The minimum distance (λmin) in the u-space, directs towards the most likely failure point, or theso-called design point (x*). Following failure points were obtained, Table 6.7.

parameter elastic-brittle model, λmin=3.78Failure point x*

elastic-plastic model, λmin=4.46Failure point x*

FVewdfcft

147.4 kN218.0 mm590.52 mm545.18 mm3.02 N/mm2

0.30 N/mm2

137.24 kN216.75 mm572.93 mm517.23 mm2.86 N/mm2

0.22 N/mm2

Table 6.7: Failure points, based on the minimum distance found in the DARS analyses

Although these are the most likely failure points, of course other less likely combinations canlead to failure too. The most likely failure point is not found when a single parameter is varied,see Table 6.6, λADI,min results. As can be seen from Table 6.7, it is a specific combination of therandom variables that lead to the minimum distance in the u-space. This is in contradiction withthe findings of Waarts (Waarts, 2000), who states that for most structural problems, the minimumdistance (λmin) is found during the ADI procedure (λmin = λADI,min). This seems not to be the case.Overall, it is observed from the outcome of the random samples that following directions in theevolution of the random variables may lead to failure: FV: � ; e: ; w:� ; d:�; fc:�; ft:�

Similar conclusions can be drawn from the sensitivity analysis in the failure point u*. Thedirection cosines in the u-space are given for the different models above, Table 6.8. Thesensitivity values are also given for the resulting Response Surface as found from the DARSanalysis, between brackets. The parameters that influence the system in a positive sense havea positive sensitivity. The others, such as the force FV and the eccentricity e, that have a negative

168

effect on the system reliability, have a negative sign. The larger the influence, the higher thevalue. Main parameter of course is the eccentricity. The sensitivity of the tensile strength is nearzero, having a negligible influence on the structural problem at hand.

model αFv αe αw αd αfc αft

elastic-fractured

-0.15(-0.23)

-0.81(-0.75)

0.08(0.12)

0.48(0.49)

0.30(0.37)

0.00(0.06)

elastic-plasticfractured

-0.11(-0.35)

-0.83(-0.76)

0.06(0.17)

0.50(0.41)

0.22(0.31)

0.00(0.06)

Table 6.8: Sensitivity analysis - ULS

6.3.3. Ultimate Limit State - checkpoints according to EC1 - level IThe level I reliability according to EC1, annex A (EC1, 1994) and illustrated in Section 3.5, iscalculated for the eccentric loaded masonry column. Assume that the preset target reliabilityequals the real outcome: βT=3.66. The limit state function should be checked at severalcheckpoints. The checkpoints are composed by attributing an importance factor (α) to the load(S) and resistance (R) variables. In case of one resistance and one dominant load variable - R-S-problem - the limit state function is to be checked in the point (-0.8βT, 0.7βT). In this examplethe eccentricity is a load variable. Its sensitivity, Table 6.8: αe = [-0.83;-0.75], indeed is veryclose to the values preset in EC1. In case of several load and resistance variables, an additionalreduction factor of 0.4 is used for non-dominating variables. The checkpoints for non-dominating load and resistance variables thus become: (0.4×-0.8βT, 0.4×0.7βT). In case it is notclear which load variable is the dominating load, all combinations are to be checked. In thisanalysis it is observed that the eccentricity is the dominating load. For the resistance, thethickness as well as the compressive strength could be the dominating variable, leading to twocheckpoints, Table 6.9. Based on the importance factors (α) and the preset target reliability (βT),the checkpoints can be calculated, according to (EC1, Table A.3):

(6.10)( )Rd TRd TV

= −

= − = <

µ αβ σ

µ αβ σ µ

(Normal probability distribution)

(Lognormal probability distribution, for Vexp . )0 2

In case of a lognormal distribution function and V>0.2, Eq. 5.21 should be used.

169

variable importance factors (α), case 1

checkpoint 1 importancefactors (α), case 2

checkpoint 2

FV [kN]e [mm]w [mm]d [mm]fc [N/mm2]ft [N/mm2]

(S) 0.4×0.7(S) 0.4×0.7(R) -0.4×0.8(R) -0.4×0.8(R) -0.8(R) -0.4×0.8

154.35214.05564.86564.862.4100.177

(S) 0.4×0.7(S) 0.4×0.7(R) -0.4×0.8(R) -0.8(R) -0.4×0.8(R) -0.4×0.8

154.35214.05564.86512.163.3560.177

gULS elastic-brittle elastic-plastic

-0.24 0.21

-0.97 0.03

Table 6.9: Checkpoints for verification on level I, according to EC1, for βT = 3.66.

Based on the outcome of the ultimate limit state function, the structure would be consideredunsafe in case the elastic-brittle model is used, although the level III analysis reveals that thepreset reliability is reached. This is mainly because - in contradiction with the direction cosines(αi) as listed in the sensitivity analysis, Table 6.8 - the square root of the quadratic sum of theimportance factors is larger then unity (1.23). The importance factors are conservatively chosento be used for any type of structural system. Thus they are not optimized for this specificstructure. The structures’ safety is therefore underestimated in most cases. From the outcomeof the ultimate limit state functions listed in Table 6.9, it is concluded that the main resistanceparameter is the thickness, having a more negative outcome.

6.3.4. Serviceability Limit State - probabilistic evaluation on level IIIAlthough the reliability index for the ultimate limit state meets the target reliability index, theserviceability limit state is most certainly violated. In other words, the chance that the masonryis cracked is very high. This can be seen from Table 6.10 and 6.11. The first table summarizesthe results based on a non tension material (NTM) model. This means that cracks will occurfrom the moment the vertical force is beyond the mid-third borders, see Section 4.2. The secondmodel, Table 6.11 accounts for the limited tensile strength of the masonry.

Method β pf YLSFE V(pf)

exact valueMVFMFORM NLPQLFORM RFLSSORMMC (σh =1)DSDARSMCARS

-1.96-1.96-1.96-1.96-1.96-1.97///

0.980.980.980.980.980.98///

/82317505.000///

/////0.5 %///

Table 6.10: SLS-Summary of results - reliability analysis - fractured elastic model (NTM)

With the DS, DARS and MCARS procedures, no results are obtained, as the root finding

170

algorithm does not work when the center point is already in the failure domain. This is the casein this example:

(6.11)g E d eSLS NTM, = −� � = − = −6

6006

150 50

For the case of the NTM model, the exact reliability index can be obtained analytically, using Eq.2.38. The random variables are normally distributed and the limit state function is linear:

(6.12)βσ σ

=−� � −

+= − −

+ ���

��

= −E d e

e d

60

50 0

25 306

1962

6

22

2.

Method β pf YLSFE accuracy

exact valueMVFMFORM NLPQLFORM RFLSSORMMC (σh =1)DSDARS (λadd=var)MCARS+VI (λadd=3.00)MCARS+VI (∆g,add=var)

0.520.460.460.460.520.520.560.520.480.55

0.30

0.300.280.300.320.29

/82317505.00087016029028

/////V(pf)= 1.5%V(pf)= 9%, σ(β)=0.42 V(pf)= 9%, σ(β)=0.42 V(pf)= 9%, σ(β)=0.42 V(pf)= 9%, σ(β)=0.42

Table 6.11: SLS- summary of results - reliability analysis - ft£0

Reliabilitymethod

SLS-reliability analysis - elastic model ft£0

DARS (λadd=var) MCARS+VI (λadd=3.00) MCARS+VI (∆g,add=var)

λADI,min

λmin

βRS,SORM

λadd,fin//∆g,add

σh,finRS-modelF/Fα

0.710.590.5070.45/pure quadraticK0

0.71/0.5283.001.0pure quadraticK0

0.71/0.5327.3 10-4

1.0full quadraticK10-15

Table 6.12: Numerical details DARS and MCARS+VI

For the serviceability limit state it can be seen that the effect of the limited tensile strength isvisible and significant in the obtained reliability index. Qualitative, this could already be

171

concluded from Figure 6.4. Indeed, the difference in eccentricity for the first crack to appear israther large. This difference is significant and is quantified by means of the reliability index.Although, in both cases the target reliability (βT,SLS = 2.1) index according to EC1 (EC1, 1994)is not met. The sensitivity analysis reveals the importance of the tensile strength with regard tothe serviceability limit state, Table 6.13. The first model, based on a NTM model, of course isonly function of d and e, as these parameters define the mid third area. The other parameters donot influence the result. When the tensile strength is accounted on, it becomes an importantparameter. Remark that in neither case the compressive strength is influencing the result as itis no part of the limit state function.

model αFv αe αw αd αfc αft

elastic-NTM

0 -0.98(-0.98)

0 0.20(0.20)

0 0

elastic-ft£0

-0.16(-0.16)

-0.70(-0.70)

0.08(0.08)

0.30(0.30)

0 0.62(0.62)

Table 6.13: Sensitivity analysis - SLS

6.3.5. Time dependencyAssume that the parameters are time-dependent (t). The eccentricity e(t) tends to increase dueto settlements of the foundation, the compressive strength fc(t) decreases due to deteriorationprocesses and the thickness d(t) and width w(t) of the column decrease because of weatheringprocesses. For example, a linear degradation model is chosen for each of these parameters withan arbitrary degradation rate. From the obtained response surfaces (DARS) the influence can beestimated easily, Figure 6.5. For comparison, some data are added based on a reliability analysis(DARS) on the original structural model.

Based on Figure 6.5, the decrease in reliability can be calculated as a function of time if thedegradation process is known. The difference between the real outcome and the outcome basedon the estimated response surface increases with increasing time. This is mainly because theresponse surface is optimal for the original center point (mean values at t=0). Because of thedegradation processes, the center point shifts further away from the original center point. It canonly be used for a first rough estimate.

This information can be useful for a maintenance plan of the structure (Melchers, 1999). Whena lower threshold value is set equal to βT = 3.00, then, depending on the material model used,repair is required after 44 or 57 years (reliability based on the outcome of the real LSF).

172

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0 20 40 60 80 100Time (t) [years]

β

gULS- elastic-fractured checkpointsgULS-RS-elastic-fractured modelgULS: elastic-plastic checkpointsgULS: RS-elastic-plastic model

fc(t)=fc,0+a1*t; a1=-0,002N/(mm2.year)e(t)=e0+a2*t; a2=0,0004 mm/yearw(t)=w0+a3*t; a3=-0,002 mm/yeard(t) =t0+a3*t; a3=-0,002 mm/year

Figure 6.5: Influence of deterioration - time dependency

6.3.6. ConclusionsThe impact of a more accurate material model for masonry is clearly visible in the resultingreliability index or failure probability. Not only for the ultimate limit state (elastic-brittle Îelastic-plastic): β== 3.66 Î 4.06 or pf = 1.26 10-4 Î 2.45 10-5, but also for the serviceability limitstate (NTM Î ft£0): β== -1.96 Î 0.52 or pf = 0.98 Î 0.30. The safety almost increases with anorder of magnitude. Better knowledge thus might lead to less conservative designs. Or, becauseof the better understanding of the structural behavior, the better correspondence of the modelwith reality, the resulting reliability indices converge closer to the real values. Elder structures,designed using the elastic-brittle model or even NTM-models, tend to be on the conservativeside.

Although the strong non-linearity in the ultimate limit state function, for which a 2nd orderpolynomial is not an optimal functional form, the DARS and MCARS+VI procedures are ableto estimate the failure probability very accurately. Because the limit state function is availableanalytically, traditional reliability methods such as FORM/SORM have a comparable efficiency.

Using probabilistic techniques, the importance of each (random) variable towards failure isquantified. This information is not available from a limit states design concept. In that, overallinfluence factors are used to account for the different variables governing the problem at hand.It has been demonstrated that these are conservative and thus might lead the wrong decision.

When the resistance or load variables are subjected to deterioration processes, the effect ofdeterioration on the resulting reliability can be calculated. When the analysis is based on theestimated response surface, it should be always reminded that the outcome will depend on thequality of the estimated response surface and that the response surface is optimal around the

173

Figure 6.6: Arches - Failure modes and safety factors αg and αS [taken from (Smars, 2000)]

original center point. In those cases, the outcome based on the response surface can only berelied on as a rough estimate.

6.4. Safety of masonry arches

6.4.1. Problem DefinitionTo evaluate the arch’s safety for a given set of parameters, the thrust line method is used(Heyman, 1969, 1985). It is a Limit Analysis (LA) method using the equations of equilibriumand the resistance characteristics of the materials. In the case of arches, it is supposed that: • blocs are infinitely resistant,• joints resist infinitely to compression, • joints do not resist to traction and • joints resist infinitely to shear.

These hypotheses are certainly restrictive: the material(s) used to construct the arch do not respectthem strictly. It was nevertheless shown that - under normal circumstances - they are reasonable(Heyman, 1985). The limits of this theory are discussed elsewhere (Smars, 2000). Also from theformer application, Application 2, Table 6.8, it can be seen that these assumptions are acceptable.Most important parameters are the eccentricity (e: αe=-0.83) and thickness or height of the arch(d: αt=0.50). The compressive strength (fc: αfc=0.30) is of secondary importance.

An additional objective of this example is to present a methodology able to make a reasonable

174

assumption of the reliability of a masonry arch, without a profound material investigation thatwould require destructive tests. Based on geometrical data, as obtained from an in site survey,it should be possible to give an indicative value of the arch’s safety. Based on the outcome, itmight be decided to provide more information concerning the material properties.

The safe theorem for an arch (Heyman, 1966; Der Kooherian, 1952), Section 4.3, alreadysuggests that more then one solution can be found. Indeed, any thrust line that satisfies the safetheorem requirements, is sufficient to ensure stability. It even does not have to be the actual lineof thrust. Accounting for dead weight only, Figure 6.6 (a, b) or an additional external force,Figure 6.6 (c,d), two extreme lines of thrust can be found, in which three hinge points are formed.Because of its degree of static undefinedness, three hinges are required to transform the structureinto a static structure. Hinges are formed when the thrust line becomes a tangent line with theintrados or the extrados of the arch.

Typical failure modes associated with the limit situations are represented in Figure 6.6. In (a)a safe situation is shown. Two extreme lines of thrust are drawn, both having three points inwhich the thrust line is tangent with the intrados or extrados. As the thickness of the archdecreases (b), the two extremes converge. Finally, when the arch is on the limit between stableand unstable, these two extreme lines of thrust converge to coincident lines. This results in 5hinges (because of symmetry), which transforms the structure into a mechanism of collapse.

Similarly, when an external force is applied (c). As long as the actual thrust line remains withinthe two extreme thrust lines with 3 hinges, the structure is in a safe state. When the externalforce increases (d), both extreme thrust lines start to converge and finally coincide, resulting in4 hinges. Again a mechanism of collapse is formed.

Given these hypotheses, it can be shown (Der Kooharian, 1952) that an arch is stable if a thrustline, remaining entirely inside its shape, can be found. Analytical expressions relating parametersto stability are not available for generic situations. The code Calipous was developed to computenumerical estimates (Smars, 2000). In particular, it can determine safety factors.

The geometrical factor of safety (αg) is defined as the minimal multiplicative factor on the archthickness allowing an internal thrust line to be found. The static factor of safety (αs) is definedas the maximum multiplicative factor on external forces allowing an internal thrust line to befound.

As long as the safety factors exceed one, the structure is in the safe region, the arch is stable. Ifone of the safety factors is smaller than one, the structure is in the unsafe region, the arch isunstable. This results in the following limit state functions:

(6.13)( ) ( )( ) ( )

g

gg

S

1

2

1

1

X X

X X

= −

= −

α

α

6.4.2. Evaluating the safety of an existing archAs an example, the global failure probability of a masonry arch, Figure 6.7, will be determined.

175

FV~LN(750,(150)²)

d~N(0.16,(0.02)²)

50 blocsr=r0+drr0~N(2.5,(0.02)²)dr~N(0,(0.02)²)

Figure 6.7: Fictitious existing masonry arch

Besides, possible actions in case of insufficient safety will be discussed. A fictitious examplewas designed to illustrate the method.

The situation (geometry and forces) is chosen so as to lead to a low geometrical safety factor (αg= 1.23). The arch being not far from instability, the uncertainties on the geometry start to playa role. The geometry indeed is not exactly known. Following parameters are chosen to approachthe possible variations:• the mean radius of the arch r0,• its thickness d,• the 2nd order deviation with respect to a perfect circular arch dr and• an eccentric vertical load FV, applied at the most critical position.

These parameters are a translation of the uncertainties that might occur during a survey of theexisting arch, Table 6.14. In practice, one needs to choose them carefully. If the important onesare neglected, the evaluation of the failure probability will yield to non-significant results.Indeed, in some cases, other factors could be of importance (local geometrical faults forinstance).

Randomvariables

Parameter meanvalue (µ)

standarddeviation (σ)

cov [%] Probabilitydensity functionPDF

inner radiusthicknessradius deviationvertical force

x1 = r0 [m]x2 = d [m]x3= dr [m]x4 = FV [N]

2.500.160.0750

0.020.020.02150

0.812.5/20

NNNLN

Table 6.14: Random variables and their parametersThe force application point on the arch was chosen to minimize the geometrical and static safetyfactors on the perfect geometry. For this particular example, the standard deviations of thedifferent random variables were chosen arbitrarily but in a real case these should be deduced

176

from the measured geometry accounting for the survey technique(s) and for the force, usingguidelines (Melchers, 1999).

The mean value of these parameters results in the following safety factors:• geometrical factor of safety: αg = 1.23,• static factor of safety: αs = 2.39.

6.4.3. Reliability analysis - DARS and MCARS+VI An interface between the DARS/MCARS+VI optimization algorithm and the Calipous programwas made to allow for automatic processing (Schueremans et al., 2001a, 2001b; Smars, 2000).Each time the limit state function needs to be evaluated, an automatic call to the external programCalipous is made. The resulting safety factors (αg and αS) are written to an output file. Thecommand is returned to the reliability procedure and the output of the LSF is used to update thereliability values and response surface. The process is illustrated in a global flow chart of theanalysis, Figure 6.8.

The standard deviation (σ(ß)) or the coefficient of variation (V(ß)) on the reliability index areused as a criterion to stop the sampling process, see Section 2.3: σ(ß)=0.15 or V(ß)=0.05. As theprocessing time for an analysis using Calipous is tolerably small (15 up to 20 seconds peranalysis), the focus was not put on optimal time efficiency. However, it is marked where thepreset accuracy is reached and the corresponding number of LSFE are listed. The overall resultsare summarized in Table 6.15. The computer time (CPU: Pentium 667 MHz, RAM: 128 MB)and number of limit state function evaluations (LSFE) are listed.

Reliability method β pf YLSFE(CPU-time)

accuracy

DARS (λadd= 3.0)DARS (λadd=var)MCARS+VI (λadd=3.00)MCARS+VI (∆g,add=var)

1.261.221.251.30

0.110.120.110.10

149 (50 min)43 (17 min)45 (18 min)23 (12 min)

V(pf)= 33%, σ(β)=0.15V(pf)= 33%, σ(β)=0.15V(pf)= 33%, σ(β)=0.15V(pf)= 33%, σ(β)=0.15

Table 6.15: Results of reliability analysis for the initial parameters

To reach the preset accuracy σ(β)=0.15, 371 samples and 149 LSFE are required for the DARSprocedure with λadd = 3.0. This position is marked on Figure 6.9. Remark that these 371samples did only require 149 direct limit state function evaluations or calls to the Calipousprogram, although this is higher than 15n = 60 as preset by Waarts (Waarts 2000). For the othersamples, the estimated response surface was accurate enough to replace time-consuming directlimit state function evaluations. After 500 samples, the standard deviation on the reliability indexσ(β)=0.14. Coarse directional sampling would require approximately 2 hours and 10 minutesfor 500 samples. Using a variable additional distance, DARS (λadd = var), the required numberof limit state functions was reduced significantly to 43. A similar conclusion can be drawn fromMCARS+VI (λadd=3.00), where 45 LSFE are sufficient. Using a variable distance in theoutcome space, MCARS+VI (∆g,add=var), even reduces the number of LSFE to 23.

177

DARSStep 1. ADI search the roots λ of the LSF for the principal directions in the u-space (n=4): 1 0 0 0 -1 0 0 0 0 1 0 0 0 -1 0 0 0 0 1 0 0 0 -1 0 0 0 0 1 0 0 0 -1

Step 2. Fit a Response Surface through the data X, Y

Step 3. Iterative procedure (i) untill required accuracy is reached Perform crude DS on the RS ui = [u1 u2 u3 u4] ||u||=1 If λ i,RS < λmin+λadd

Calculate pi(LSF)=χ2(λ i,LSF,n) Update the RS with (Xi,Yi) Update λadd

Else Calculate pi(RS)=χ2(λ i,RS,n)

Root-findingalgorithm in x-space[2.5 0.16 0.0 750]….[2.5 0.10 0.0 750][2.5 0.14 0.0 750][2.5 0.134 0.0 750]….

[x1 x2 x3 x4]

Calculateoutcome LSF[0.23]…[-0.45][0.05][0.01]→λ=1.278…

[yi]→=λi,LSF

Calipous

X Y

Xi Yi

generate data files:input.dat; arc.dat ;arc.chareturn command toCalipous

generate outputfiles:output.txtreturn commandto Matlab

Figure 6.8: Global flow chart of the automatic processing between DARS and Calipous

Some numerical details are listed in Table 6.16. These give a proper insight in the analysis andstructural behavior as well. The minimum distance to the origin (λmin = 1.28) is already foundin the Axial Directional Integration procedure (ADI), Step 1 in the DARS and MCARS+VIprocedure, see Figure 6.8. A pure or full quadratic response surface fit the data well, as theadditional distance, based on the error between real outcome and response surface outcome, isrelatively small. The final additional distance equals: λadd,fin = 0.11, which is significantly lowerthen the arbitrary distance, preset by Waarts (Waarts, 2000). Thus, a significantly lower amountof LSFE will be performed. The fact that the estimated response surfaces are valid, is confirmedby the small ratio’s F/Fα. The same counts for the additional distance in the outcome space:∆g,add,fin=0.016., in case of MCARS+VI (∆g,add=var). A significantly lower number of limit statefunction evaluations is performed, compared to MCARS+VI (λadd=3.00), where the additionaldistance is kept constant.

Reliabilitymethod

DARS(λadd= 3.0)

DARS(λadd=var)

MCARS+VI(λadd=3.00)

MCARS+VI(∆g,add=var)

178

λADI,min

λmin

βRS,SORM

λadd,fin//∆g,add,fin

σh,finRS-modelF/Fα

1.28==λADI,min = 1.281.223.00/pure quadratic10-19

1.28==λADI,min = 1.281.220.11/full quadratic10-17

1.28/1.223.001.00full quadratic10-17

1.28/1.170.0161.00pure quadratic10-17

Table 6.16: Numerical details DARS and MCARS+VI

For the response surface, a polynomial of second order was used. As the form of the limit statefunction is not known beforehand, its functional form has to be estimated. For most structuralproblems a polynomial of first or second order is appropriate. In other cases a more complexrelation has to be searched for (Montgomery, 1997). To illustrate that the resulting responsesurface estimates the reliability of the structure quite well, the probability of failure is calculatedby means of a FORM/SORM analysis using the final estimate of the response surface as limitstate function in stead of the original problem (βRS,SORM), Table 6.16.

The most likely failure point (with minimum distance to the origin: λmin = 1.28) in the u-spaceequals u*=[0,-1.28,0,0]. The corresponding failure point in the x-space reads: x* =[2.5m,0.1344m,0m,750N] and coincides with the case of minimum thickness in the arch, Table6.17. The sensitivities based on the different reliability methods are listed too. The two othermodes in the ADI-procedure that lead to failure are less likely, as can be seen from their distanceto the origin in the u-space (λ i):• u*=[0,-1.28,0,0] Õ x* = [2.5m,0.1344m,0m,750N] with λmin = 1.28,• u=[0,0,10.68,0] Õ x = [2.5m,0.16m,0.21m,750N] with λ = 10.68,• u=[0,0,0,4.32] Õ x = [2.5m,0.16m,0m,1730N] with λ = 4.32.

Parameter r [m] d [m] dr [m] FV [N]

failure point (x*) 2.50 0.1344 0.0 750

sensitivities αr αd αdr αFv

DARS (λadd= 3.0)DARS (λadd=var)MCARS+VI (λadd=3.00)MCARS+VI (∆g,add=var)

-0.03-0.03-0.04-0.04

0.940.970.960.97

-0.11-0.11-0.11-0.14

-0.32-0.24-0.25-0.20

Table 6.17: Sensitivity analysis

From Table 6.17 it is seen that the thickness (d) of the arch is the main parameter, having thelargest influence in reaching failure.

The 95% confidence interval for the reliability index (ß) and global failure probability pf are: 95%CI(ß) = [0.97;1.47] or 95% CI(pf)= [0.07;0.17]. The evolution of the reliability index as afunction of the number of simulations is presented in Figure 6.9, Figure 6.10 and Figure 6.11.Upper- and lower boundaries of the 95% confidence interval are shown in dashed line. The

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Increased thickness:d: µµµµ=0.21 m; σ=0.02 m; N=192#LSFE=126, β=3.72, pf=1.0 10-4

Increased accuracy:d:µ=0.16 m; σσσσ=0.005 m; N=273LSFE=122,β=3.44,pf=2.9 10-4

Initial survey:d: µ=0.16 m; σ=0.02 m; N=371#LSFE=149, β=1.26, pf=1.0 10-1

Number of Samples N

β

95% Confidence interval β

βT=3.7

Figure 6.9: DARS (λadd= 3.0) outcome - Reliability β versus number of Samples N

results based on DARS (λadd = 3.0) and DARS (λadd=var) represented in Figure 6.9 and Figure6.10. Figure 6.11 and Figure 6.12 present the results for the MCARS+VI (λadd=3.0) andMCARS+VI (∆g,add=var) procedures respectively.

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Increased thickness: d: µµµµ=0.21 m; σ=0.02 m; N=198#LSFE=45; β=3.69, pf=1.1 10-4

Increased accuracy: d: µ=0.16 m; σσσσ=0.005 m ; N=484#LSFE=34, β=3.55, pf=1.9 10-4

Initial survey: d: µ=0.16 m; σ=0.02 m ; N=52#LSFE=43; β=1.22, pf=1.1 10-1

β

95% Confidence interval β

βT=3.7

Number of Samples NFigure 6.10: DARS (λadd= var) outcome - Reliability β versus number of Samples N

Increased thickness: d: µµµµ=0.21 m; σ=0.02 m; N=813#LSFE=34; β=3.83, pf=0.6 10-4

Increased accuracy: d: µ=0.16 m; σσσσ=0.005 m ; N=4254#LSFE=60, β=3.43, pf=2.9 10-4

Initial survey: d: µ=0.16 m; σ=0.02 m ; N=384#LSFE=45; β=1.25, pf=1.1 10-1

β βT=3.7

Number of Samples NFigure 6.11: MCARS+VI (λadd=3.00) outcome - Reliability β versus number of Samples N

.

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Increased thickness: d: µµµµ=0.21 m; σ=0.02 m; N=386#LSFE=23, β=3,67, pf=1.2 10-4

Increased accuracy: d:µ =0.16 m; σσσσ=0.005 m; N=3677LSFE=56, β=3.46,pf=2.7 10-4

Initial survey: d: µ=0.16 m; σ =0.02 m; N=413#LSFE=23, β=1.30, pf=1.0 10-1

Number of Samples N

β95% Confidence interval β

βT=3.7

Figure 6.12: MCARS+VI (∆g,add=var) outcome - Reliability β versus number of Samples

6.4.4. Upgrading the reliability index - different possibilitiesThe previous results indicate that the structure does not meet the target reliability index accordingto the European Standard Eurocode 1 (EC1, 1994): ßT = 3.7. To meet the safety requirements,two options can be taken: (1) perform a more accurate survey of the arch or (2) strengthen thestructure. In both cases it is interesting to determine the most critical parameters with respect tothe failure probability in order to maximize the efficiency of the action(s) taken. Of course, ifthe safety was considered sufficient, then nothing has to be done.

The direction cosines of the final estimate of the response surface at the failure (or design) point(Table 6.17: [αr, αd, αdr, αFv]) are measures of the relative influences of these parameters withrespect to the reliability index β. Here, the thickness of the arch is the most important parameter(αd = 0.94). The gain of a more accurate survey of the thickness can be evaluated quickly usingthe response surface. Thus, if the standard deviation could be limited to σ(d) = 0.005 m (insteadof the original value of 0.02 m), the response surface directly gives a first estimate of theprobability of failure: ß=3.42 or pf=2.9 10-4. To confirm this value a, a new DARS/MCARS+VIanalysis is performed to update this first estimate. These data are added in Table 6.18 and onFigure 6.9 (DARS λadd=3.0), Figure 6.10 (DARS λadd= var), Figure 6.11 (MCARS+VI λadd=3.0)and Figure 6.12 (MCARS+VI ∆g,add=var). The resulting reliability index ß = 3.44-3.55 isslightly higher than the value estimated based on the response surface. It is interesting to see thatthe reliability index ß = 3.44-3.55 almost equals the target reliability index ßT = 3.7, accordingto EC 1 (EC1, 1994). If it is decided that this is an acceptable safety level, further consolidationwould no longer be required, leaving the structure maximally unaffected, in its authentic state.

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At this point, the importance of the choice of parameters again must be stressed. The parameter“time” could be very important: the abutments may settle inducing changes in the globalgeometry. If this is the case, the initial model might no longer cover the real structural behaviorand the outcome would no longer be valid. If so, monitoring of these settlements can be used toensure the validity of the chosen set of parameters.

reliability analysis - increased accuracy on the thickness d: (µ = 0.16 m, σ = 0.005 m)

Reliability method β pf YLSFE (CPU-time)

accuracy

DARS (λadd= 3.0)DARS (λadd= var)MCARS+VI (λadd= 3.0)MCARS+VI (∆g,add=var)

3.443.553.433.46

2.1 10-4

1.9 10-4

2.1 10-4

2.7 10-4

122 (41 min)34 (13 min)60 (22 min)56 (21 min)

V(β)=0.05V(β)=0.05V(β)=0.05V(β)=0.05

reliability analysis - increased mean thickness d: (µ = 0.21 m, σ = 0.02 m)

Reliability method β pf YLSFE (CPU-time)

accuracy

DARS (λadd= 3.0)DARS (λadd= var)MCARS+VI (λadd= 3.0)MCARS+VI (∆g,add=var)

3.723.693.833.67

1.0 10-4

1.1 10-4

0.6 10-4

1.2 10-4

126 (52 min)45 (17 min)34 (14 min)23 (12 min)

V(β)=0.05V(β)=0.05V(β)=0.05V(β)=0.05

Table 6.18: Results of reliability analysis for increased accuracy or thickness

If it is decided not to perform a more accurate survey for reasons of accessibility, time orexpenses, then a consolidation would become necessary. One could for instance look for therequired mean thickness of the arch to achieve the standard reliability of βT=3.7 preset in EC1.This is most probably not a realistic reinforcement project, but it exemplifies the use of themethod to calibrate interventions. Again the response surface leads to a first estimate: µ(d)=0.21m (βRS=3.63). The reliability index increases to: β=3.67-3.83 or: pf=0.6-1.1 10-4, see Figure 6.9(DARS λadd = 3.0), Figure 6.10 (DARS λadd = var), Figure 6.11 (MCARS+VI λadd= 3.0) andFigure 6.12 (MCARS+VI (∆g,add=var). A summary for the different reliability methods DARSand MCARS+VI is added in Table 6.18. The numerical details again are summarized in Table6.19.

183

increased accuracy thickness d(µ = 0.16 m, σ = 0.005 m)

increased mean thickness d(µ = 0.21 m, σ = 0.02 m)

Reliabilitymethod

DARSλadd=var

MCARS+VIλadd=3.0

MCARS+VI∆g,add=var

DARS λadd=var

MCARS+VIλadd=3.0

MCARS+VI∆g,add=var

λADI,min

λmin

βRS,SORM

λadd,fin//∆g,add

σh,finRS-modelF/Fα

4.43 3.603.420.05/p-q10-17

4.43/3.383.001.98f-q10-18

4.43/3.350.091.98f-q10-17

3.783.783.691.60/f-q10-17

3.783.783.673.002.13f-q10-17

3.783.783.720.0252.14f-q10-16

Legend: RS-model: p-q for pure quadratic and f-q for full quadraticTable 6.19: Numerical details DARS and MCARS+VI

Following remarks can be made for the case of the increased accuracy:• the minimum distance is no longer found during the ADI procedure. The minimum

distance in the ADI procedure corresponds with a maximum tolerable vertical load equalto FV=1767 N. Because of a higher accuracy in the thickness, this failure point becomesless likely. The three modes in the ADI-procedure that lead to failure are:

u=[0,-5.30,0,0] Õ x = [2.5m,0.1344m,0m,750N] with λ = 5.30,u=[0,0,10.68,0] Õ x = [2.5m,0.16m,0.21m,750N] with λ = 10.68,u=[0,0,0,4.43] Õ x = [2.5m,0.16m,0m,1767N] with λ = 4.43.

• The minimum distance λmin = 3.60, corresponds with a combination of lower thicknessand higher vertical load. The failure point (x*) equals: x* =[2.49 m, 0.146 m,0.02m,1096 N]. The thickness no longer is the dominating variable. The direction cosinesat the failure point in the u-space equal: α = [-0.10,0.60,-0.27,-0.74]. The importanceis shifted towards the vertical load. Due to a lower standard deviation on the thickness,the original failure point is no longer the most likely failure point. Another failure pointbecomes more likely.

• The reliability procedure itself decides which model represents the LSF the best. Theuser does not have to interfere in this choice. It is based on minimum error, Section3.3.4.

• From the reliability index based on the response surface, it can be concluded that again,the outcome of the resulting response surface very well coincides with the real implicitLSF.

For the case of the increased mean thickness similar remarks can be made:• The minimum distance λmin = 3.78, corresponds with the minimum distance found in the

ADI procedure: λADI,min = 3.78, and thus corresponds with a minimum thickness equalto 0.134 m. The most likely failure point (x*) is the same as the original problem.Because the mean thickness of the arch is increased, the distance to the origin in the u-space - and thus reliability - is increased. Because of the higher thickness, only twomodes in the ADI-procedure, leading to failure, remain:

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u*=[0,-3.78,0,0] Õ x* = [2.5m,0.1344m,0m,750N] with λmin = 3.78,u=[0,0,0,9.61] Õ x = [2.5m,0.16m,0m,4936N] with λ = 9.61.

• The reliability procedure uses a full quadratic response surface to describe the implicitLSF.

• The reliability index, based on the final estimated response surface, is nearly the same ason the real LSF. Again, a good match is established. This can also be seen from the lowratio F/Fα.

6.4.5. ConclusionsThis example illustrates the practical applicability of the DARS and MCARS+VI procedure incase of implicit limit state functions. When an automatic interface is available between thereliability analysis and the procedure that gives the outcome of the LSF, the system reliability canbe calculated in a closed loop without further interference of the calculator.

As there are two implicit failure modes, DARS and MCARS+VI are very suitable procedures.In case FORM/SORM would have been used, the solution would converge to a componentreliability representing the reliability of a single failure mode. A system analysis would beneeded afterwards to retrieve the system reliability.

The Limit Analysis code Calipous enables to calculate safety factors (ag and aS), using a NTMmodel, based on a few geometrical parameters. As these are gathered in a geometrical survey,a first estimate of the reliability can be calculated.

In case the present reliability is judged insufficient, the sensitivity analysis in the failure point(u*) allows to assess the effect of possible future interventions. In this case, the thickness of thearch is the main variable that influences the structures’ safety. To increase the reliability, twopossibilities remain: increase of the mean value of the thickness, or decrease of its variance. Thelatter means a more accurate survey, if possible. The former leads to an impact on the structureitself. The response surface can be used to evaluate the effect of this measures quickly. Anincreased accuracy upgrades the reliability from 1.26 to 3.44-3.55, which almost equals thecommon target reliability index. If it is judged sufficient, the monument remains intact, keepingits authenticity. If not, an increased thickness (d = 0.21 m) results in an upgrade of the reliabilityindex up to 3.69-3.83. In both cases an additional reliability analysis is performed to give a moreaccurate value for the reliability index. Remark that in both cases the estimate based on theresponse surface is rather excellent. This will not always be the case.

The use of a variable additional distance λadd, again confirms to be a workable improvement inpractice. The number of LSFE remains significantly lower compared to the traditional DARSmethod, using λadd = 3.0. In this case the gain in computer time is more important, since a LimitAnalysis using Calipous requires about 20 seconds. The benefit is larger as far as the estimatedresponse surface is a good representation of the real LSF. The fact that the resulting additionaldistance is limited, λadd,fin = 0.05-1.60, confirms this assumption. It is further confirmed by thefact that the estimated reliability outcome, based on the response surface, is very close to theoutcome based on the original LSF. This is taken advantage of by using a variable λadd.

Finally, the reliability procedure itself chooses which functional form from a family of low orderpolynomials, is best to describe the implicit limit state function. In some of the analyses this is

185

a pure quadratic function, in other, a full quadratic polynomial is preferred.

A similar reflection can be made for MCARS+VI. Adding an additional distance in the outcomespace (∆g,add=var), leads to a significant improvement. The response surface very well estimatesthe real structural behavior. For MCARS, the response surface is based on all LSFE, not onlythe roots, as this is the case for the DARS procedure. Final additional distances in the outcomespace are very small: ∆g,add,fin= 0.016-0.09.

When DARS is compared with MCARS+VI, both methods seem to be suitable for this type ofproblems. No significant difference in efficiency can be retrieved from this example. Wheneverthe whole structural system can be described accurately with a low order polynomial, MCARSseems to be more efficient. This is because all LSFE are used to estimate the response surface.In case of DARS, intermediate results are omitted. This is a loss of information.

6.5. Masonry Shear Walls

6.5.1. Problem definitionThe interest in shear walls did increase the last 10 years because of their multi-modal failurebehavior (masonry crushing, tensile failure, shear failure) that was difficult to predict or even tosimulate using finite element analyses. In the Netherlands a global research program was setup(CUR 171, 1994; Van der Pluijm, 1992; Vermeltfoort and Raaijmakers, 1992; Vermeltfoort etal., 1993; Vermeltfoort and Janssen, 1996). Main target was to provide a numerical andexperimental basis for the design of structural masonry (CUR 171, 1994). Shear walls have beentested and results were compared or used to calibrate numerical models and finite elementanalysis.

Shear walls are designed to withstand lateral loads. These originate generally from wind loads,in which case they remain relatively small in absolute value. In earthquake sensitive regions thelateral load can be induced by the earth movement. In that case the lateral loads can become thedominant loading.

To obtain sufficient similarity with reality, a non-linear finite element analysis is required. In thisexample, a macro-scale continuum model is used (Lourenço, 1996). The different failure modesthat may lead to collapse are described by the material model. Besides the non linear finiteelement model, a linear elastic and analytical model are treated, their accuracy and effect on theresults are discussed.

The overall geometry is visualized in Figure 6.13. The nominal size is w x h = 1.0 x 1.0 m,chosen for simplicity.

186

FV

FH

width = 1.00 m

heig

ht =

1.0

0 m

Figure 6.13: Shear wall geometry

Different failure mechanisms appear during collapse and, depending on the ratio between verticalpre-stressing and horizontal load, one or the other will be dominating. Overall, the failuremechanisms are shown in Figure 6.14 (Lourenço, 1996). The vertical pre-stressing force isimposed to prohibit early tensile cracking of the masonry panel. As the lateral load is imposed,the shear wall initially tends to rotate, which is prohibited by the boundary conditions: thevertical displacements are locked. Initially, Figure 6.14 (a), both tensile and compressive stressesappear at the edges of the masonry. The shear stresses are still equally distributed over the fulledge. By further increasing the lateral loads, the tensile strength of the masonry is reached andtwo cracks appear in the diametrical opposite corners, Figure 6.14 (b), where the tensile stressesare induced in the panel. From now on, stresses in the masonry remain concentrated in the non-cracked region. With further increase of the lateral load, the cracks grow, so that thecompression and shear stresses are concentrated in the two opposite corners. Suddenly, adiagonal crack will appear due to shear in the mid region, partly in the bed or vertical joints,partly in the bricks due to shear forces, Figure 6.14 (c). Finally, depending on the amount of pre-stress, crushing may occur due to concentrated compressive stresses in the two opposite corners.

187

(a) (b) (c)

Nor

mal

stre

sses

Shea

r stre

sses

Figure 6.14: The failure modes and stress distribution of a masonry shear wall, schematicrepresentation according to (Vermeltfoort and Raijmakers, 1992)

As masonry only behaves linear elastic for low stresses, a linear elastic model is onlyrepresentative in that domain. Its use is limited to the prediction of the first tensile cracks. It canbe used for a kind of serviceability limit state control. Reaching the ultimate strength, thecalculated stresses will not correspond with the actual stresses when a linear elastic model isused. The uncertainty that has to be applied on the mathematical model to correct for that errorgoverns the whole problem, as already presented elsewhere (Schueremans et al., 1999c). For anaccurate estimate of the safety of masonry shear walls, a non-linear finite element analysis isindispensable. The material model has to be able to simulate the stress redistribution after tensilecracking.

6.5.2. Finite element modelingThe problem of shear walls was dealt with using Diana 7.2 (Diana, 1998b) during a short termresearch period at TNO in Delft (Netherlands). This research was conducted in close cooperationwith Van der Pluijm (Van der Pluijm, 1999) and Waarts (Waarts, 2000). The most promisingaspect of the collaboration was the fact that in the framework of the doctoral research of Waarts,

188

a probabilistic shell was developed for Diana 7.2, named Probab, which was implemented in apilot version of Diana 7.2 (Waarts, 2000). This shell allows for probabilistic design of structuralsystems using finite element analysis.

Although this creates a very powerful environment, the main difficulty lies in the masonrymaterial modeling. Because of its brittle behavior, the required iterative procedure and thenumerical complexities involved, convergence problems were the main obstacle. It is wellknown that these problems can be treated in a more efficient way using a displacement controlledanalysis instead of force controlled analysis, which was confirmed in practice (Diana, 1998).Drawback for this type of problems however, was that the Probab-procedure did not providedisplacements to be defined as random variables. In the pilot version at hand, only nodal orelement forces could be defined as random variables. This necessitates the user to work force-controlled, which in the case of masonry often leads to diverging results.

To overcome this drawback, the problem was analyzed using Atena2D, a commercial finiteelement package developed for the analysis of (quasi)-brittle materials, with emphasis onconcrete (Cervenka, 2001). The material model used is presented in Chapter 4. Although anabsolute convergence could be guaranteed because of the displacement-controlled analysis, noautomatic communication between the finite element analysis in Atena2D and probabilisticprocedure in Matlab is available. Moreover, the Atena2D code does not provide this option. Therequired manual interaction stresses the importance to limit the number of finite elementanalyses. Therefore, the DARS procedure with a variable additional distance (λadd = var) waschosen.

6.5.3. Reliability analysis - ULSThe random variables used in this example are summarized in Table 6.20. The initial verticaldisplacement is a translation of the vertical dead weight on top of the wall, to prohibit earlytensile cracking. An average pre-stress of approximately 1.0 N/mm2 is assumed:

(6.14)σ εy initial y E dyh

E Nmm

Nmm, . .≈ = = × =0 000604 1642 1002 2

The coefficient of variation is taken 9% for proper weight/permanent load (Melchers, 1999). Thehorizontal force (FH) is the active horizontal load. Its value is arbitrarily chosen.

The Atena2D finite element analysis for the mean values is shown in Figure 6.15. The insetfigures of the deformed shear walls, show the damage factor of the Atena material model usedfor the analysis. It demonstrates the tensile cracking in the tips, the crushing of the masonry andthe shear failure in a shear band area.

189

Random variables Parameter mean value standarddeviation

cov[%]

PDF

S

R

x1x2x3x4x5x6x7

horizontal loadvertical displacementYoungs modulusPoissons’ ratiothickness of wallcompressive strengthtensile strength

FH [kN]dy [m]E [N/mm2]ν [-]d [m]fc [N/mm2]ft [N/mm2]

456.04 10-4

16420.190.1884.260.28

11.255.4 10-5

1400.060.0080.810.10

2598320.41936

LNLNNLNNLNLN

Table 6.20: Random variables

The vertical pre-stressing by means of an initial vertical displacement is applied in the first loadstep. The vertical displacements are locked and the horizontal displacement is increased stepwiseuntil failure. It can be seen that the shear wall withstands the average load it is subjected to,when the average material parameters are used. For the average horizontal force (FH = 45 kN),the shear wall remains in the elastic domain, no tensile cracking occurs. The maximumadmissible horizontal force equals: FH,max = 117.7 kN. The corresponding safety, seen theuncertainty on the different parameters, is subject of the reliability analysis.

For the reliability analysis, DARS (λadd=var) is used. The limit state function reads:

(6.15)gF

FULSh

h

average= − = − =,max . .1 117 7

451 162

The outcome equals 1.62 for the average values, listed in Table 6.20. The results of thereliability analysis are summarized in Table 6.21. The numerical details are listed in Table 6.22.The evolution of the reliability index as function of the number of samples is presented in Figure6.23.

Reliability method β pf YLSFE (time) accuracy N

DARS (λadd=var) 3.54 2 10-4 153 (K28 hours) V(β)=0.05 385Table 6.21: Results of reliability analysis for the initial parameters

Although the safety margin seemed to be relatively large, the reliability index does not meet thetarget reliability index (β = 3.54 < βT = 3.7). To meet the preset accuracy, V(β) = 0.05, 153 limitstate function evaluations are required. Each limit state function evaluation represents a finiteelement analysis using Atena2D. The manual handling and finite analysis itself takesapproximately 10 minutes. The total processing time is estimated to be around 28 hours. Whenan automatic interface would be available, the processing time could be reduced to the CPU timeneeded for the finite element analysis, which is about 6 minutes. The probabilistic proceduredoes only require a few seconds. The same analysis would take 17 hours and could runovernights.

190

0

20

40

60

80

100

120

140

0 1 2 3 4 5 6 7Horizontal displacement dx [mm]

Horizontal displacement for mean horizontal force

FH,max = 117,7 kN

X

Y

X

Y

dxFH

First tensilecracking

71,7

45

0,56

Figure 6.15: Horizontal force - displacement diagram for masonry shear wall analyzed for itsmean values

Reliability method DARS (λadd=var)

λADI,min

λmin

βRS,SORM

λadd,finRS-modelF/Fα

4.0553.9653.391.63pure quadratic0.6/3.87

Table 6.22: Numerical details DARS (λadd=var)

The advantage of using a variable additional distance (λadd = var) can be seen from Figure 6.23.The final additional distance is 1.63, which is significantly lower than the arbitrarily value presetby Waarts: λadd=3.0 (Waats, 2000). This results in a lower number of LSFE required. Initially,Figure 6.23, the additional distance is higher. This is because the response surface is not accurateenough as only a small number of LSFE is available. When the estimated response surfaceresults in a better estimate of the real behavior, the additional distance decreases. The evolutionof the additional distance, as function of the number of samples, is given in Figure 6.23.

191

2,00

2,20

2,40

2,60

2,80

3,00

3,20

3,40

3,60

3,80

4,00

0 100 200 300 400 500 600N

ββββ

0

2

4

6

8

10

12

14

λλλλaddβ = 3.54, V(β)=0.05, N=385 #LSFE=153, pf = 2.0 10-4

95% CI(β)

λadd=var

λadd=3.0

βT = 3.7

Figure 6.16: DARS (λadd=var) - β and λadd versus number of samples N

The minimum distance to the origin is not found yet after the ADI procedure, Table 6.22. Duringthe random sampling a closer distance to the origin is found. The most likely failure point (withminimum distance to the origin: λmin = 3.965) in the u-space equals u*=[2.32, -0.18, -1.47, 1.26,-0.99, -2.30, -0.55]. The corresponding failure point in the x-space reads: x* = [77.29 kN, 5.9210-4 m, 1436 N/mm2, 0.27, 180.05 mm, 2.71 N/mm2,0.21 N/mm2 ], Table 6.23. The only modein the ADI-procedure that leads to failure is less likely, as can be seen from its distance to theorigin in the u-space (λ i): u=[4.055,0,0,0,0,0,0] Õ x = [118.48 kN, ...] Õ with λ = 4.055. Themodel chosen, a pure quadratic polynomial, is sufficiently accurate, which can be concluded fromthe ratio of F/Fα.

The sensitivities in the failure point x* are listed in table 6.23. The only negative direction cosinecorresponds with the horizontal force applied. The vertical pre-stressing (dy) increases the safety.

Parameter FH[kN]

dy[m]

E[N/mm2]

ν=[−]

d [mm]

fc[N/mm2]

ft[N/mm2]

failure point (x*) 77.29 5.92 10-4 1436 0.27 180.05 2.71 0.218

sensitivities αFh α∆y αE αν αd αfc αft

DARS (λadd=var) -0.86 0.05 0.07 0.01 0.15 0.48 0.06Table 6.23: Sensitivity analysis

6.5.4. Reliability analysis - SLSFor the serviceability limit state, a linear elastic model can be applied. For this analysis it is

192

presumed that the serviceability limit state is violated when tensile cracks occur:

(6.16)g fSLS t y= − σ

The random variables are listed in Table 6.24.

Random variables Parameter µ σ cov [%] PDF

S

R

x1x2x3x4x5x6

Horizontal loadVertical displacementYoung’s modulusPoisson’s ratiothickness of walltensile strength

FH [kN]dy [m]E [N/mm2]ν [-]d [m]ft [N/mm2]

456.04 10-4

16420.190.1880.28

11.255.4 10-5

1400.060.0080.10

2598320.436

LNLNNLNNLN

Table 6.24: Random variables

The system behaves linear elastic until that point, Figure 6.15. To allow for automaticprocessing, an identical model is built in Calfem (Calfem, 1999) which is a general purpose finiteelement toolbox in Matlab. The results of the reliability analysis are summarized in Table 6.25.The numerical details are listed in Table 6.26. It is observed that the reliability index meets thepreset target reliability index for serviceability limit states, according to EC6: β = 2.46 > βT = 2.1.

Reliability method β pf YLSFE accuracy N

DARS (λadd=var) 2.46 7 10-3 372 σ(β)=0.15 743Table 6.25: Results of reliability analysis for the initial parameters

The minimum distance, or the failure point is already found during the ADI procedure. Itcorresponds with a maximum horizontal load: u*=[2.481,0,0,0,0,0,0] Õ x* = [72.9 kN, ...] Õwith λ = 2.481. This value agrees very well with the average horizontal load at which firsttensile cracking is observed, Figure 6.15.

Reliability method DARS (λadd=var)

λADI,min

λmin

βRS,SORM

λadd,finRS-modelF/Fa

2.4812.4812.401.01full quadratic3.5/3.85

Table 6.26: Numerical details DARS (λadd=var)

6.5.5. Reliability analysis - analytical model according to EC6Finally, the reliability is calculated according to EC6. The analytical model as proposed by SIAV177/2 is used, see Section 4.4.2. Following limit state function is applied:

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(6.17)

( )

( )g

f l dF

compression

c l d FF

shear

c c

V

c v

H

=

× × ×−

× × + ×−

�

min

cos

tan

2

1

1

α

φ

in which the length of the compressed part (lc) is calculated according to, see Figure 4.9:

(6.18)l l F hF

and bg FFc

H

V

H

V

= − = � �2 : tanα

The random variables are listed in Table 6.27.

Random variables Parameter mean value standarddeviation

cov[%]

PDF

S

R

x1x2x3x4x5x6x7x8

Horizontal loadVertical load thickness of wallcompressive strengthcohesionfriction coefficientlengthheight

FH [kN]FV [m]d [m]fc [N/mm2]c [N/mm2]tan(φ) [-]lh

451880.1884.260.500.8111

11.2516.920.0080.810.150.15//

2590.4193019//

LNLNNLNLNLNfixedfixed

Table 6.27: Random variables

From this analysis it can be seen that the resulting reliability is very low and significantly smallerthan the value found using the non-linear finite element analysis: β=1.6<3.55. The model israther conservative. This is mainly because it lacks similarity with reality, as already mentionedin Section 4.4.2. The results of the reliability analysis are summarized in Table 6.28. Thenumerical details are listed in Table 6.29.

Reliability method β pf YLSFE accuracy N

DARS (λadd=var) 1.60 5.5 10-2 78 σ(β)=0.15 243Table 6.28: Results of reliability analysis for the initial parameters

Reliability method DARS (λadd=var)

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λADI,min

λmin

βRS,SORM

λadd,finRS-modelF/Fα

1.8241.8241.670.25full quadratic0.04/3.90

Table 6.29: Numerical details DARS (λadd=var)

6.5.6. ConclusionsThis example treats the reliability analysis of a masonry shear wall for the ultimate as well as forthe serviceability limit state.

To obtain an accurate value for the resulting reliability of the ultimate limit state, the DARS(λadd=var) procedure is combined with a non-linear finite element analysis. This is necessary toaccount for the physical non-linear behavior of the masonry shear wall. For comparison, thereliability is calculated using an analytical model according to EC6 too. It is demonstrated thatthe result is conservative and underestimates the load bearing capacity of the shear wall at hand.

For the serviceability limit state, the occurrence of tensile cracks, a linear elastic analysis is used.Therefore the Matlab finite element toolbox Calfem is used. Although the ultimate limit statedoes not satisfy the target reliability index (β=3.55<βΤ=3.70), the serviceability limit state meetsthe preset target reliability index (β=2.46>βΤ=2.1).

The major advantage of Calfem as implemented in Matlab is its open program structure. Theprogram code can easily be consulted and extended for own purposes. The mutualcommunication between Calfem and the probabilistic procedure in Matlab are written in such away that interference of the user is no longer required. Its applicability is also demonstrated inAnnex A.15. In that, the reliability of a plane frame steel structure is calculated. Drawback ofthe generic finite element code Calfem is that it is not developed for the very specialized analysisof quasi-brittle materials, such as masonry. The material models available are not capable ofdescribing shear walls up to collapse. Therefore it can only be used to show the power of themethodology by means of illustrating the SLS analyses.

Atena2D on the contrary is a finite element code, developed especially for quasi-brittle materials.Because of its numerical stable behavior and because its availability at the Civil EngineeringDepartment, it was chosen to be used for the required finite element analyses. Major drawbackof this methodology was the lack of automatic communication between Atena2D and theprobabilistic analysis in Matlab. The user has to interfere for each finite element analysis.Although the action required by the user is small, it has to be repeated for each limit statefunction evaluation. This for example makes it impossible for this type of analysis to runovernights.

6.6. Conclusions

This chapter illustrates the outlined reliability procedures on several examples. Not only

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analytical problems are treated, such as grout-injection for consolidation of three-leaf masonrywalls or an eccentrically loaded masonry column. The methodology is also combined with morecomplex masonry structures such as arches and masonry shear walls. In these, physical nonlinear behavior is dealt with. With the first examples, emphasis is on historical masonry. Withthe final example, the accent is shifted towards more contemporary masonry applications.

To distinguish the safe from the unsafe region in more complex applications, an external programis used: a limit analysis Calipous in case of masonry arches, finite element analyses Calfem andAtena2D in case of masonry shear walls. If an automatic interaction is established between theexternal program (Calipous, Calfem) and the reliability procedure, a powerful tool for thecalculation of reliability values is obtained. If not (Atena2D), it emphasizes the need to minimizethe number of limit state function evaluations required. In that, the use of DARS with a variableadditional distance (λadd=var) gains interest.

The efficiency of the different reliability methods is compared. Not only for mathematicalexamples, Annex A, but also for structural applications. The MCARS and DARS methods areextensively compared with traditional reliability methods in the eccentric loaded masonrycolumn.

The use of a variable additional distance λadd, confirms to be a workable improvement in practice.Depending on the value of λadd, the number of LSFE may be significantly lower compared to thetraditional DARS method, using λadd = 3.0. The benefit is larger when the estimated responsesurface is a good representation of the real LSF. This is translated in low values for λadd,fin. Itis further confirmed by the fact that the estimated reliability outcome, based on the responsesurface, is very close to the outcome based on the original LSF. This is taken advantage of byusing a variable λadd.

A similar reflection can be made for MCARS+VI. Adding an additional distance in the outcomespace (∆g,add=var), leads to a significant improvement when the response surface very wellestimates the real structural behavior. For MCARS, the response surface is based on all LSFE,not only the roots.

The efficiency and applicability of MCARS+VI and DARS are mutually compared in detail forthe masonry arch application. Both methods seem to be suitable for the reliability analysis ofstructural systems. Whenever the whole structural system can be described accurately with a loworder polynomial, MCARS+VI seems to be more efficient. This is because all LSFE are usedto estimate the response surface. Furthermore, no complex root-finding algorithm is required.In case of DARS, intermediate results are omitted, the response surface is based on the rootsonly. This is a loss of information. The advantage of DARS is in the calculation of the roots ofthe limit state function. These is very interesting information from a structural point of view.Omitting intermediate results, it enables to fit a 2nd order polynomial in the important region only,namely for the critical situation, where gLSFK0. In case of structural behavior that can not becaptured by a 2nd order polynomial. This is a real advantage.

Finally, the reliability procedure itself chooses which functional form from a family of low orderpolynomials is best to describe the implicit limit state function. In some of the analyses this is

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a pure quadratic function, in other, a full quadratic polynomial is preferred.

For the case of an eccentrically loaded masonry column and shear walls, reference is made to thereliability according to EC6. In these applications, attention is not only paid on the ultimate limitstate, but also on the serviceability limit state.

The effect of different material models on the resulting reliability index is treated. It is observedthat elder models are more conservative and underestimate the actual safety values. Modernmaterial models that include plastic behavior or make use of the physical non-linear behavior ofmasonry, show closer similarity with reality. In the examples treated, this leads to higherreliability values. In general, the obtained reliability is believed to approximate the real reliabilityto a greater extent. This effect is quantified explicitly for the case of the eccentrically loadedmasonry column and for shear walls.

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7 General conclusions and further research

7.1 General conclusions

A general methodology is established to obtain objective safety values for existing structures.This methodology is mainly illustrated on unreinforced historical masonry structures. Of course,dealing with masonry structures only affects the material model and structural model. Thegeneral methodology is applicable to other building materials as well when the correspondinginformation concerning the material and structural model are available, Annex A.15. To obtainobjective safety values, a level III method is put forward. All variables can be treated as randomvariables accounting for the present uncertainties.

Existing target safety values are reviewed. An alternative is proposed that enables the culturalheritage value to be included in the case of historical buildings, Section 1.1.5.

A comprehensive overview of traditional reliability methods that are available for calculatingsafety values is given, Chapter 2. Their capability in obtaining global failure probability valuesaccording to a level III approach is evaluated and illustrated on an academic example. For the different sampling procedures, the required accuracy is commented. An adapted criterionis proposed, depending on the actual reliability level, Section 2.3.

Recently developed combined methods are treated in Chapter 3. These methods combinetraditional reliability procedures such as Directional Sampling and Monte Carlo Sampling, withan Adaptive Response Surface. The response surface methodology is treated and improvementsare added to the DARS procedure, increasing its overall efficiency. Main improvements addedby the author are:• an additional variable distance is introduced (λadd=var). This distance distinguishes the

important from the non-important region. The variable distance is based on the error ofthe roots in the standard normal space (u-space), between the real limit state function andthe estimated response surface, Section 3.3.4;

• the procedure itself chooses the functional form from a family of second orderpolynomials that fits the data best, Section 3.3.4;

• an F-test is used to check whether or not the resulting response surface is acceptable,Section 3.3.2.

Also a promising alternative is worked out in detail, MCARS+VI, in which Monte CarloVariance Increase is combined with an Adaptive Response Surface, Section 3.4. Besides addingvariance increase (VI) to limit the number of samples, two versions are compared, similar toDARS. In the first an arbitrary distance λadd=3.0 is maintained, in the other a variable distancein the outcome space ∆g,add is proposed. The latter is based on the error in the outcome spacebetween real outcome and response surface outcome.

With respect to both combined reliability methods, the main conclusions are:• The efficiency and applicability of DARS and MCARS+VI are checked on a variety of

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limit state functions of varying complexity. Therefore 15 academic examples are addedin Annex A. 14 of them have already been dealt with using DARS (λadd=3.0). The gainof using DARS (λadd=var) and MCARS+VI is pointed out. Overall, the efficiency ofDARS and MCARS+VI are comparable and depend on the capability of the usedresponse surface in capturing the real system behavior. For more complex applicationsthe efficiency of MCARS+VI is believed to be less than DARS. This is caused by thefact that MCARS+VI uses all LSFE to construct the response surface. For complexstructures a low order polynomial might no longer be efficient to capture the overallbehavior of the structure, whereas it might be sufficient to represent the roots. The latterprocedure is used in DARS.

• In other cases, when the overall behavior can be estimated accurately using a low orderpolynomial, MCARS+VI will be more efficient (for example: eccentric loaded masonrycolumn, Section 6.3). This is because MCARS+VI uses all LSFE to estimate theregression coefficients of the response surface. Then, omitting the intermediate resultsin DARS, is omitting information.

• Advantage of the DARS procedure is that it finds the roots of the limit state function,where MCARS+VI only returns a positive or negative value of the limit state function.The roots of the limit state function contain interesting information with respect to thefailure modes of the structure. On the other hand, MCARS+VI does not require anycomplex root finding algorithm. Therefore, the procedure remains very simple andstraightforward.

• If the limit state function is available analytically, most methods are convenient in findingthe global failure probability, also the traditional methods, Chapter 2. When this is notthe case, and moreover, when it requires substantial amount of computational effort, acombination of Directional Sampling or Monte Carlo with an Adaptive Response Surfaceare preferable.

The masonry material models that are used for the limit state formulations in the differentapplications, are reviewed and rephrased, Chapter 4. These are required to distinguish the safefrom the unsafe region. In several cases the border between safe and unsafe can be expressedanalytically, in other cases more complex numerical models are required. Therefore, analyticalas well as numerical finite element models are treated. For the analytical models focus is onmasonry in compressive regime. Because centric compression will hardly ever be the case,eccentric loading is treated in detail. Besides non tension material model (NTM), the influenceof a limited tensile strength and plastic behavior are treated. Arching is described subsequently.The design models available in EC6 are rephrased. Partial safety factors are omitted to allowthese models to be used as the limit state formulation in a probabilistic approach. Some of these(models for lateral loading and shear walls) are rudimentary and might lead to a biased reliabilityindex. To obtain greater similarity with reality, numerical models are the obvious way. For thenumerical modeling of masonry, focus is on finite element modeling. Although a detailed micro-model would deliver more accurate results, a macro-model is used for reasons of efficiency. Theglobal anisotropic continuum model in the elastic, plastic and post-peak behavior, proposed byLourenço, is treated. The limitations with respect to available finite element codes - Diana 7.2and Atena2D - are listed.

Up to now, statistical information on masonry is limited. Chapter 5 addresses the experimentalresearch. From the models described in Chapter 4, a variety of material properties is derived.

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When a probabilistic evaluation procedure is aimed at, the probability distribution of thesematerial properties needs to be determined. Different sources that are available to gatherexperimental data are reviewed. The research focused on three levels: the components brick andmortar, the composite masonry and the relationship between components and composite. Inorder to obtain significant comparisons of test results and a complete dataset, only one typemasonry (brick, mortar and bond) is used. For the same purpose, the number of test data is takensufficiently high. The main findings are: • For the components brick and mortar, the probability distribution function has been

determined for the main material properties (fc, ft, E), based on experimental results.Different test setups have been used to acquire the brick compressive strength, which ledto different outcomes. Based on hypothesis tests these differences were judgedsignificant. Besides the sample size, the anisotropic behavior of the bricks is the maincause.

• For the masonry composite, the probability distribution functions of the main materialparameters are no longer based on experimental research solely. The required samplesize restricts the number of samples that is practically acceptable. Besides a limitednumber of tests on cores, pillars and small wallets, numerical relationships have beenexplored. For the uni-axial compressive state, use was made of the formula provided byEC6. Stochastic extension of this formula delivered acceptable results to predict theprobability distribution function of the masonry compressive strength from the compositevalues. For the Young’s modulus, the homogenization technique proved to be a validalternative.

• For masonry in shear regime, test results on cores and small wallets were supplementedwith literature values. This allowed to provide the material parameters’ distributionfunction of Coulombs’ friction law for this type of masonry.

• For masonry in tensile regime, the distribution function of the masonry tensile strengthwas determined experimentally using cores drilled from couplets.

• Masonry in a multi-axial stress state was treated based on triaxial testing. For the hybridmortar used, a significant strength increase is demonstrated, even for low confinement(σ3/σ1-ratio’s). Furthermore, the stress-strain diagrams for different confinement ratio’sshow a change in material behavior from quasi-brittle to elastic-plastic with increasingσ3/σ1-ratio’s. These large plastic deformations already take place at low σ3/σ1-ratio’s,e.g.: σ3/σ1=0.05. For the heterogeneous masonry samples, the number of successful testswas too limited to retrieve a triaxial yield criterion spanning the hole σ1-σ2=σ3 stressplane. Due to oil leakage in the triaxial cell or reaching the maximum confining pressure,it was not possible to derive reliable test results for high σ3/σ1-ratio’s. As a consequence,the apparent anisotropic behavior of masonry could not be confirmed from thisexperimental test setup. Biaxial test results in literature on small masonry panels builtfrom comparable full brick masonry, do show a limited degree of anisotropic behavior.For masonry samples, a similar strength increase for low confinement ratio’s could bequantified. A transition from quasi-brittle to elastic-plastic behavior was observed forconfinement ratio’s exceeding 0.4.

• The historical aspect puts extra complexity towards obtaining (experimental) data for thematerial properties. It is stressed that obtaining representative values is hard in practice.Extra complexity and uncertainty in case of existing buildings come from the fact that itwill never be possible to determine the material properties from the building itself to theirfull extent. In many cases, information on the workmanship and in site conditions can

200

hardly be quantified.• The influence of the test setup and measurement method are not negligible, such as test

bank stiffness, Section 5.3.4. Moreover the error made can not always be quantified veryprecisely, which is experienced with the measurements of the lateral deformations in caseof triaxial testing, Section 5.8.4. Besides, it is wishful thinking that for each safetyassessment of an historical building, the amount of test results will be available as it isthe case in this study. Therefore, databases should be more reliable and become a validalternative for test results. In addition, numerical models tend to be more representativefor the real material behavior. The price we pay is the extra information needed. Oftenmore parameters are required for that purpose, which puts an increased emphasis onexperimental research.

In the applications, Chapter 6 and Annex A, a variety of problems is dealt with, combining thedifferent elements described in the former Chapters. These applications confirm the applicabilityof the methodology in practice. In some cases the limit state function is known analytically(Annex A.1-A.14, grout-injection for consolidation of three-leaf masonry walls, Section 6.2, oran eccentric loaded masonry column, Section 6.3), in other cases each limit state functionevaluation requires a limit analysis (safety of masonry arches, Section 6.4) or a non linear finiteelement calculation (Shear walls, Section 6.5). In the latter, not only linear elastic materialbehavior is treated (plane frame steel structure, Annex A.15), but physical non linear behavioris taken into account. With the first examples (Section 6.2-6.4), emphasis is on historicalmasonry. With the final example (masonry shear walls, Section 6.5), the accent is shifted towardsmore contemporary masonry applications. Following conclusions are drawn:• To distinguish the safe from the unsafe region in more complex applications, an external

program is often used: a limit analysis Calipous in case of masonry arches, finite elementanalyses Calfem and Atena2D in case of masonry shear walls. If an automatic interactioncan be established between the external program (Calipous, Calfem) and the reliabilityprocedure, a powerful tool for the calculation of reliability values is obtained. If not (e.g.:Atena2D), it emphasizes the need to minimize the number of limit state functionevaluations required. In that, the use of DARS with a variable additional distance(λadd=var) gains interest.

• The corresponding reliability was calculated, with the analytical limit state functionaccording to EC6. In these, the proposed limit state functions were used, after strippingof (partial) safety factors. Attention was not only on the ultimate limit state, but also onthe serviceability limit state. The effect of different material models on the resultingreliability index is treated. It is observed that elder models or simplified analyticalmodels often used by standard codes are more conservative and underestimate the actualsafety values. Modern material models that include plastic behavior or make use of thephysical non-linear behavior of masonry, show closer similarity with reality. In theexamples treated, this leads to higher reliability values. In general, the obtainedreliability is believed to approximate the real reliability to a greater extent. This effectis quantified explicitly for the case of the eccentrically loaded masonry column and shearwalls.

7.2 Further research

In this section, possible subjects for further research are listed. Some of them are closely relatedto the subject of this work, others are related to adjacent research fields or a combination of both.

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• Introducing neural networks in stead of low order polynomials for the response surfacemay reduce the required number of limit state functions significantly. Using low orderpolynomials to fit the response surface might not be an optimal choice for structuralproblems. When the response surface does not reproduce the real behavior accurately,this is compensated by the reliability analysis by introducing a large additional distanceλadd. This results in a higher number of limit state function evaluations. Using neuralnetworks, no functional form is preset (Simon, 1999; Martin, 1999; Saad, 1998). It canbe looked at as a ‘universal approximator’. The basic weighting functions are veryflexible and adapt well to any type of functional behavior. It has no restriction on shape,degree of complexity, singularity or function type. Initially, when a small number oflimit state function evaluations is available, the knowledge of the neural network will belimited. During the analysis the number of limit state functions increases and thus theinformation and knowledge in the neural network improves. The outcome will convergeto the real limit state function outcome, not limited by a preset functional form.Preliminary results on masonry structures are very encouraging (Shin and Pande, 2001).This will enhance the efficiency of MCARS+VI as it uses all the LSFE and not only theroots. DARS could be extended in a similar way and include all LSFE to fit the responsesurface and estimate the roots. Its efficiency is believed to become comparable.

• Developing a graphical user interface (GUI) would improve user friendly applicability.In the framework of this doctoral research, a significant number of Matlab m-files hasbeen written. During implementation, the readability is taken care of by usingcomprehensible variable names, adding comment lines and adopting a structured layout.But for the moment, the use of the different probabilistic procedures is still on thecommand line level. The limit state function still has to be defined in an m-file as wellas the matrices with random variables and their parameters. A GUI could combine thesealgorithms in a probabilistic toolbox (Marchand, 1996; Mathworks, 1997). The userfriendliness would become comparable to commercial packages such as Comrel/Sysrel(RCP, 1997). Although this is no scientific research as such, it is a prerequisite to havethe methodology used in engineering practice. In other research fields, such as systemidentification, it has led to successful implementations (Macec Toolbox: Peeters, 2000).

• Linking the probabilistic procedures implemented in Matlab, with finite element codesthat are implemented in Matlab, or enable automatic communication with Matlab wouldallow for a wide range of structural applications to be dealt with. Femlab for exampleis a general purpose finite element code, that is mainly written in Matlab (Femlab, 2001).The open m-file code allows the user to inspect algorithms, make changes to them, or addnew features. For the moment, the finite element code does not focus on quasi–brittlematerials such as masonry. Physical non linear behavior is limited to plasticity. Finiteelement codes dealing particularly with the quasi-brittle behavior of masonry (such asDiana, Atena2D) that have an open program structure or are implemented in Matlab arenot available at this moment.

• During the doctoral research, the reliability procedures are mainly illustrated on masonryexamples. Because of their complex structural behavior, masonry applications are notstraightforward. In most cases the reliability procedure is not the problem, but thedistinction between the safe and unsafe region is. Because the numerical method

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required, is the limiting factor, the examples treated remain relatively simple. Morecomplex structures can be dealt with when progress is made on the combination of bothfields. When dealing with structures built with less complex materials, such as steel, thelatter is not a problem. Annex A.15 for example, contains the analysis of a plane framesteel structure, using the finite element code Calfem (Calfem, 1999). Problems withlarge dimensions and a large number of random variables are handled successfully withease. The reliability of a cable-stayed bridge with 67 random variables is calculatedwithin 4 hours (Waarts, 2000). Therefore, also in other application fields, related tostructural restoration, the reliability methodology might offer interesting perspectives.For example, the reliability of externally bonded CFRP laminates on concrete beams, isa very promising subject. When a reinforced concrete beam is strengthened usingexternally bonded CFRP laminates, different failure modes may occur: bending failure,shear failure, anchorage failure (delamination, plate end shear...). Recent developmentshave led to analytical models for these different failure modes, that very well coincidewith experimental results (Ahmed 2000, Matthys 2000, Brosens 2001). Which failuremode is most likely to occur will depend on different parameters: the loads applied, thedesign of the CFRP strengthening and the material properties of the concrete beam. Allthese variables have their uncertainty, such as: the surface tensile strength of the concrete,the future service load, the workmanship at gluing, the angle of the carbon fibers or theexperienced model uncertainty. In case of strengthening, it is expensive to increase thestructural reliability. A methodology that is capable of predicting the reliability of thestrengthened beam accounting for the different failure modes and actual uncertainties,would mean a step forward in optimal design meeting preset reliability values.

• In the continuous assessment of monuments, the reliability analysis on itself is only onelink of the chain. To allow a global reliability based maintenance plan to be setup,interdisciplinary research combining reliability, optimization and life-cycle costing (LevelIV, see Chapter 2) is required. The step towards reliability based assessment is being setin bridge maintenance in the USA (Frangopol and Kong, 2001). For the conservationof cultural built heritage, first attempts in this direction are made too (COASCO, 2001).Several challenges are identified, such as the role of maintenance in reliability and therelation between structural monitoring and reliability in continuous assessment. Finally,this should result in the establishment of formal criteria for optimal cost-based andreliability-based decision making, the development of codified rules for definingreliability states and simplified procedures for assessment of reliability states. Of coursethis is a slow process. Indeed, it is not sure that reliability based assessment can offerlarge cost savings even though it is a vastly more rational method. Moreover, there is alot of inertia caused by different factors such as unfamiliarity with the probabilisticapproach and the lack of reliable databases on performances and costs of interventionson existing structures.

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225

Annex A - Academic Examples

The examples that are treated in this annex overlap for the greater part with the examples covered by Waarts(Waarts, 2000) and are partly copied from his analyses. As some improvements are added to the DARS reliabilitymethodology, leading to DARS (λadd=var) and MCARS+VI (λadd=3.0 and ∆g,add=var), their influence on the resultsis evaluated in this annex. The new values are added in bold. For ease of comparison, the same accuracy as presetby Waarts is maintained (based on V(pf)). The outline is taken similar for all the examples. Following items aregiven: limit state function, table with random variables and their parameters, the results based on different reliabilitymethods, some numerical details and intermediate results and finally some comment or further explanation. Forsome examples the analytical outcome is calculated as the solution is straightforward and illustrates the numericalimplications.

These examples, annex A.1-A.14, cover a wide variety of possible limit state functions of varying complexity. Theywere developed to compare the different reliability methods with respect to some preset criteria (Waarts, 2000):1. multiple critical points,2. noisy boundaries,3. unions and intersections,4. space dimension,5. probability level,6. strong curvatures in the limit state function,7. no roots in the axis direction.

The criteria that are influencing the reliability procedure for a specific example are listed in Table A.1.

The final example, A.15, calculates the reliability of a plane frame. In that, the combination and automaticcommunication between the reliability method and the finite element code Calfem (Lund, 2000) which is alsoimplemented in Matlab, result in a stochastic finite element method.

The effect of the changes into the reliability methods can be evaluated based on these examples. It can be seen thatthe altered DARS method is handling the problem similar as the original DARS method (Waarts, 2000), withrespect to the above mentioned criteria. The altered methodology increases the efficiency, as the distance λadd isno longer chosen arbitrarily, but based on the estimated response surface error, Section 3.3.4. Furthermore, thefunctional form of the response surface is determined during the analysis, based on minimum error, Section 3.3.4.In that, an optimal choice is made. The final functional form used, is mentioned for each of the analyses. Ofcourse, for those examples with an initial functional form that is covered within a second order polynomial, thechosen response surface is identical to the original limit state function, as this will lead to a minimum error.

For the MCARS methodology, variance (VI) increase is added. The sampling variance (σh²) is optimized basedon the obtained results, Section 3.4.1. This leads to a more or less optimal sampling and results in a methodologywith efficiency comparable to the DARS method. To distinguish the important from the unimportant region, twooptions are foreseen. First, similar to DARS, an arbitrarily distance λadd = 3.0 is proposed. Secondly, similar toDARS (λadd = var) a variable distance in the outcome space is defined, ∆g,add. The final value of these distancesare listed in the Table A.*.3, dealing with the numerical details. The larger these values, the worse the responsesurface is capable of capturing the real behavior. This leads to a higher number of limit state function evaluations.

In case the limit state function has a simple functional form - that can be represented by a 2nd order polynomial -the response surface overlaps with the initial limit state function. In those cases the additional distance convergesto zero: λadd = 0.

In the other cases, the additional distance is a measure for the validity of the response surface. It decreases whenmore limit state function evaluations become available and the response surface becomes a better estimate of thereal limit state function. Because the regression coefficients result from a least squares estimate, the mean valueof the error εroot converges to zero. This means that no systematic bias is present between the response surface andthe real limit state function.

226

This information can also be seen from the validity test of the response surface. The validity of the response surfaceis calculated according to the methodology outlined in Section 3.3.2. The F-value is compared with a presetsignificance level (Fα), in which α=0.05. The model is accepted if F<Fα. Whenever the response surface fits thedata well, low ratio’s are obtained. When this is not the case, higher values are recorded. In any case, the validationof the RS is checked and the user is warned if the model validity is insufficient.

For DARS, the starting value is taken λadd=3.0 as proposed by Waarts. After the number of roots found amountsnroot = 2×n+1 (n the number or random variables), λadd is updated with Eq. 3.27 each time a new root is found. Thisis done to avoid that by coincidence a perfect overlap is obtained during the axial directional integration (ADI: step1 and 2 in the DARS analysis).

Similar, for MCARS+VI (∆g,add = var), the variable distance in the outcome space is applied whenever nLSFE=2×n+1 LSFE are available.

As can be seen from the academic examples, the SORM results based on the Response Surface are better when theResponse Surface estimate is better (and the limit state function is nearly quadratic). The smaller λadd, the betterthe estimate of the real LSF, the more reliable the SORM estimate based on the RS.

The efficiency of the different sampling procedures, expressed by the number of LSFE (nLSFE), using an adaptiveresponse surface are summarized in Table A.1.

efficiency(nLSFE)example

criteria DARS(λadd=3.0)

DARS(λadd=var)

MCARS+VI(λadd=3.0)

MCARS+VI(∆g,add = var)

A.1 / 18 9 41 5

A.2 2 271 41 246 31

A.3 4 38 13 32 9

A.4 4,6 221 40 36 36

A.5 4,6 188 86 86 82

A.6 7 47 16 18 13

A.7 1,4,5 160 65 762 61

A.8 3,6,7 225 19 46 15

A.9 3 55 29 11 39

A.10 3 135 138 12 92

A.11 2,3,4,6 240 85 558 18

A.12 1,3 175 85 224 169

A.13 1,3 127 314 314 314

A.14 1,3 51 51 79 192Table A.1: Summary of efficiency comparing the different sampling procedures on an adaptive response surface

227

Annex A.1 - Standard R-S problem

Limit state function: (A.1.1)g R S= −

Random Variables mean (µ) standard deviation (σ) PDF

RS

7.02.0

1.01.0

NN

Table A.1.1: Input variables

Reliability method β # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)FORM-ARS (x-space)SORMSORM + ISMC (adsamp)MC+VI (σh=3)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

3.543.543.543.543.543.543.543.543.543.543.573.553.553.533.523.54

/679309661231410000122100189415

/////////0.020.490.490.490.470.490.49

0.520.540.54

Table A.1.2: Results for different reliability methods

Method ADI DARS(λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

4.6265.003.533.00

3.553.543.543.00/

3.543.53553.53550.00/linearKKKK0

3.52/3.533.002.66linearKKKK0

3.54/3.530.002.37linearKKKK0

Table A.1.3: Intermediate results

DARS (λλλλadd= var): As the limit state function is linear, the estimated response surface overlaps with the initial limitstate function when the roots of the principle directions are found. The number of LSFE required amounts 9. After4n+1 LSFE, no error is found between estimated values and real values based on the original problem. λaddconverges to zero, further analysis is done on the response surface. An arbitrary accuracy can be reached withoutthe expense of extra LSFE.MCARS+VI (λλλλadd= 3.0): After the ADI procedure, the standard deviation of the sampling function σh is based onthe estimated reliability index: β = 4.626. This results in σh = 3.61. At the end of the procedure the final standarddeviation of the sampling function equals: σh = 2.66.MCARS+VI (∆∆∆∆g,add= var): after 2n+1=5 MC-samples (LSFE), a perfect match is obtained from the real LSF.

228

Annex A.2: Noisy limit state function

Limit state function: (A.2.1)( )g x x x x x x xii

= + + + − − +=

1 2 3 4 5 61

62 2 5 5 0 001 100. sin

Random Variables mean (µ) standard deviation (σ) PDF

x1,x2,x3,x4x5x6

1205040

121512

LNLNLN

Table A.2.1: Input variables

Reliability method β # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)FORM-ARS (x-space)SORMSORM + ISMC (adsamp)MC+VI (σh=2)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

2.252.352.352.352.232.352.352.252.262.242.242.242.242.262.222.22

/404445210103715719731000012259252714124631

/////////0.010.250.250.250.250.250.25

0.060.060.06

Table A.2.2: Results for different reliability methods

Method ADI DARS (λadd= 3)

DARS(λλλλadd= var)

MCARS+VI (λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

1.752.672.093.000.87

2.242.542.193.00/

2.262.572.350.02/linearKKKK0

2.25/2.353.001.1linearKKKK0

2.22/2.351.4 10-3

1.1linearKKKK0

Table A.2.3: Intermediate results

DARS (λλλλadd= var): As the limit state function is almost linear, the estimated response surface nearly overlaps withthe initial limit state function F/FαK0. The noise term is captured by the constant term that equals 3.8 10-3. Thisis a small value that has negligible influence on the resulting failure probability or reliability index.MCARS+VI (λλλλadd= 3.0): After the ADI procedure, the standard deviation of the sampling function σh is based onthe estimated reliability index: β = 1.75. This results ins σh = 0.87. At the end of the procedure the final standarddeviation of the sampling function equals: σh = 1.10.MCARS+VI (∆∆∆∆g,add= var):after 2n+1=31 MC-samples (LSFE), a perfect match is obtained from the real LSF.∆g,add is of the order of magnitude of the noise on the LSF. Annex A.3: R-S problem with quadratic term

229

Limit state function: (A.3.1)g R S= − 2

Random Variables mean (µ) standard deviation (σ) PDF

RS

11.01.5

1.00.5

NN

Table A.3.1: Input Variables

Reliability method β (pf) # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)FORM-ARS (x-space)SORMSORM + ISMC (adsamp)MC+VI (σh=3)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

3.463.473.473.473.473.473.473.463.473.473.503.433.433.48 (2.5 10-4)3.41(3.25 10-4)3.45

/121914346762431910000267603813329

/////////0.020.490.490.490.370.280.28

0.080.080.08

Table A.3.2: Results for different reliability methods

Method ADI DARS (λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

3.403.633.53

3.433.483.763.00/

3.433.47====

=ββββLSF = 3.470.0/pure quadraticKKKK0

3.41/====

=ββββLSF = 3.473.002.59pure quadraticKKKK0

3.45/====

=ββββLSF = 3.4702.61pure quadraticKKKK0

Table A.3.3: Intermediate results

DARS (λλλλadd= var): MCARS+VI (λλλλadd= 3.0): σh,0 = 2.57; σh,fin = 2.59 MCARS+VI (∆∆∆∆g,add= var): after 2n+1=9 MC-samples (LSFE), a perfect match is obtained from the real LSF.

230

Annex A.4: Limit state function with 10 quadratic terms

Limit state function: (A.4.1)g R Sii

i= −

=

2

1

10

Random Variables mean (µ) standard deviation (σ) PDF

RSi

0.50.2

0.10.1

NN

Table A.4.1: Input Variables

Reliability method β # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)FORM-ARS (x-space)SORMSORM + ISMC (adsamp)MC+VI (σh=2)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

2.983.203.203.202.983.203.202.972.962.983.082.902.902.962.942.97

/65617457414161516371000028681132221403636

/////////0.110.390.390.390.390.390.39

Table A.4.2: Results for different reliability methods

Method ADI DARS (λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

2.293.822.67

2.903.643.523.00/

2.963.64====

=ββββLSF = 3.200.0/pure quadraticKKKK0

2.94/====

=ββββLSF = 3.203.001.14pure quadraticKKKK0

2.97/====

=ββββLSF = 3.20=01.17pure quadraticKKKK0

Table A.4.3: Intermediate results

DARS (λλλλadd= var): MCARS+VI (λλλλadd= 3.0):σh,0 = 1.02; σh,fin===1.14=MCARS+VI (∆∆∆∆g,add= var):36 samples are needed to find the roots in the ADI procedure. Then a perfect match ofthe real LSF is found as outcome.

231

Annex A.5: Limit state function with 25 quadratic terms

Limit state function: (A.5.1)g R Sii

i= −

=

2

1

25

Random Variables mean (µ) standard deviation (σ) PDF

RS

11.01.5

1.00.5

NN

Table A.5.1: Input Variables

Reliability method β # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)FORM-ARS (x-space)SORMSORM + ISMC (adsamp)MC CrudeMC+VI (σh=1.5)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

2.632.922.842.842.522.942.942.522.632.982.632.742.652.652.642.642.64

/8413616467457575413141000010000065012540188868682

/////////0.27/0.310.310.310.310.310.31

Table A.5.2: Results for different reliability methods

Method ADI DARS (λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

1.793.472.33

2.653.473.173.00/

2.643.47====

=ββββLSF = 2.840.0/pure quadraticKKKK0

2.64/====

=ββββLSF = 2.843.01pure quadraticKKKK0

2.64/====

=ββββLSF = 2.840.01pure quadraticKKKK0

Table A.5.3: Intermediate results

DARS (λλλλadd= var): MCARS+VI (λλλλadd= 3.0):σh=1, minimum valueMCARS+VI (∆∆∆∆g,add= var): 82 samples are needed to find the roots in the ADI procedure. Then a perfect match ofthe real LSF is found as outcome.

232

u1

u2

g(u1,u2)=0

Figure A.6.1: LSF g

Annex A.6: Convex failure domain

Limit state function: (A.6.1)( ) ( )g u uu u

= + − −+

2 5 0121 2

2 1 2. .

RandomVariables

mean(µ)

standarddeviation (σ)

PDF

u1,u2 0 1 NTable A.6.1: Input Variables

Reliability method β # LSFE V(pf) σ(β)

ExactFORM standard FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)SORMSORM + ISMC (adsamp)MC+VI (σh=1.5)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

2.632.502.502.502.612.502.632.632.632.632.632.612.522.572.64

/8794847143141000051320847161813

////////0.020.320.320.320.320.320.32

0.130.130.13

Table A.6.2: Results for different reliability methods

Method ADI DARS (λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

Q

Q

2.88

2.612.502.503.00/

2.522.50====

=ββββLSF = 2.500.0/full quadraticKKKK0

2.57/====

=ββββLSF = 2.503.001.95full quadraticKKKK0

2.64/====

=ββββLSF = 2.500.01.95full quadraticKKKK0

Table A.6.3: Intermediate results

DARS (λλλλadd= var): MCARS+VI (λλλλadd= 3.0): σh,0 = 2.02; ση,φιν===1.95MCARS+VI (∆∆∆∆g,add= var): 13 samples are needed to find the roots in the ADI procedure. Then a perfect match ofthe real LSF is found as outcome.

233

Annex A.7: Oblate spheroid

Limit state function: (A.7.1)g R Sii

i= −

+=

2

1

10

1 10

Random Variables mean (µ) standard deviation (σ) PDF

RSi

100

/1

fixedN

Table A.7.1: Input Variables

Reliability method β # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)SORMSORM + ISMC (adsamp)MC CrudeMC+VI (σh=2)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

1.103.333.313.311.933.21*0.971.261.101.101.071.061.051.051.10

/20418748213922/19161310000010000016821701606576261

////////0.15/0.090.090.090.090.090.09

Table A.7.2: Results for different reliability methods (*: error message)

Method ADI DARS (λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

1.053.3246.4

1.063.32/3.00/

1.053.32====

=ββββLSF = 3.30.0/pure quadraticKKKK0

1.05/====

=ββββLSF = 3.33.001.0pure quadraticKKKK0

1.10/====

=ββββLSF = 3.30.001.0pure quadraticKKKK0

Table A.7.3: Intermediate results

DARS (λλλλadd= var): MCARS+VI (λλλλadd= 3.0):σh = 1.0 (minimum value).=The sampling variance used by Waarts for MC+VI, σh= 2.0,is not an optimal choice, as the reliability index is rather low. Because of the low reliability index, most samplesare generated in the important regionMCARS+VI (∆∆∆∆g,add= var): 13 samples are needed to find the roots in the ADI procedure. Then a perfect match ofthe real LSF is found as outcome.

234

u1

u2

g(u1,u2)=0

Figure A.8.1: LSF g

Annex A.8: Saddle surface

Limit state function: (A.8.1)g u u= −3 1 2

RandomVariables

mean(µ)

standarddeviation (σ)

PDF

u1, u2 0 1 NTable A.8.1: Input Variables

Reliability method β # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)SORMSORM + ISMC (adsamp)MC CrudeMC+VI (σh=3)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

2.342.862.452.452.562.502.582.582.582.322.342.262.352.342.272.34

/164736620334140910000100000385299225194615

///////////0.250.250.250.250.25

Table A.8.2: Results for different reliability methods

Method ADI DARS (λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

Q

Q

Q

2.352.450.03.00/

2.342.45====

=ββββLSF = 2.450.0/interactionKKKK0

2.27/====

=ββββLSF = 2.453.001.73interactionKKKK0

2.34/====

=ββββLSF = 2.450.001.77interactionKKKK0

Table A.8.3: Intermediate results

DARS (λλλλadd= var): Because no roots are found on the principal directions, see ADI in Table 3.8.3, a full factorialdesign is performed (using ff2n(n).m in Matlab (Mathworks, 2000)). Waarts (Waarts, 2000) uses random samplingin this case. The outcome is similar. In practice often no roots are found in the principal directions, see chapter5. Applications. A combination of different parameters leads to failure. This is accounted for by the checkpointsin EC, with different weights for the variables. MCARS+VI (λλλλadd= 3.0):MCARS+VI (∆∆∆∆g,add= var): 15 samples are needed to find the roots in the ADI procedure. Then a perfect match ofthe real LSF is found as outcome.

235

Annex A.9: discontinuous limit state function

Limit state function: (A.9.1)g. R S

.R SR S

=− + −

−�

∀ ≥∀ <

0 50 5

Random Variables mean (µ) standard deviation (σ) PDF

RS

155

2.50.5

NN

Table A.9.1: Input Variables

Reliability method β # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)FORM-ARS (x-space)SORMSORM + ISMC (adsamp)MC CrudeMC+VI (σh=2.0)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

3.83*6.60**3.853.846.60*3.823.833.843.943.843.823.943.79

//7//111312/10000100000032533355291139

/////////0.02/0.540.540.540.540.540.54

Table A.9.2: Results for different reliability methods (*: error message)

Method ADI DARS (λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

3.663.903.83

3.843.833.823.00

3.853.833.830.14/linear0.005/4.00

3.94/3.933.002.99linear0.95/3.86

3.79/3.930.512.87linear0.5/3.82

Table A.9.3: Intermediate results

DARS (λλλλadd= var): This limit state function has a functional form which is not covered by the preset 2nd orderpolynomial that is applied for fitting a response surface. As no intermediate results are used, this does not causeany problem. A linear model is sufficient to predict the roots easily, leading to the correct results.MCARS+VI (λλλλadd= 3.0) and MCARS+VI (∆∆∆∆g,add= var): In case of MCARS+VI, the outcome of each limit statefunction is used to construct a response surface. More difficulty is observed in estimating the correct outcome.Here the difference is seen in the estimated RS. In case of MCARS+VI the global behavior is estimated. The fitis not good. In case of DARS, only the behavior in the critical region is estimated, which is very well approximatedby a linear behavior. This is illustrated in Figure A.12.1. In there the problem is transformed to a single normal

236

Z=R-S

fZ(z)

gZ gRS

g

ββββ=3.824

Figure A.9.1: Discontinuous limit state function

variable Z=R-S.In doing so, the problem can be solved analytically:

(A.9.2)g . R S.

R SR S

Z R S g . Z.

ZZ

= − + −−�

∀ ≥∀ <

= − → = − +−

���

∀ ≥∀ <

0505

0505

00

In here, the normal variable Z has following parameters: Z~N(10,(2.55)²). From this, the reliability index can bedetermined directly:

(A.9.3)( )[ ] [ ] ( )p P g z P zf Zz

z= ≤ = ≤ =

−�

�� � = −0 0 25

0 25.

.Φ Φ

µσ β

From this equation, the reliability index β can be calculated:

(Α.9.4Α.9.4Α.9.4Α.9.4)β µσ

=−

=z

Z

0 253824

..

237

Annex A.10: Two branches

Limit state function: (A.10.1)gx x x

x xxx

=− −−

���

∀ ≤∀ >

1 2 3

3 2

3

3

55

Random Variables mean (µ) standard deviation (σ) PDF

x1x2x3

10.00.04.0

0.51.01.0

NNN

Table A.10.1: Input Variables

Reliability method β # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)FORM-ARS (x-space)SORMSORM + ISMC (adsamp)MC+VI (σh=3.0)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

5.03/4.002.21***4.844.51*5.035.155.015.005.004.825.07

//244769//29253/1000027877281351381292

/////////0.060.570.570.570.570.570.57

0.120.120.12

Table A.10.2: Results for different reliability methods

Method ADI DARS(λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

5.576.005.37

5.004.704.713.00/

5.005.124.843.34/full quadratic2.4/3.8

4.82/4.993.003.10full quadratic1.9/3.8

5.07/5.861.073.27full quadratic0.6/3.8

Table A.10.3: Intermediate results

DARS (λλλλadd= var): the response surface model is not a very good fit of the original problem, as the ratio F/Fα israther large. λadd=3.34 > 3.00, which is interesting to see. Thus, the arbitrarily value preset by Waarts is notsufficient in all examples. MCARS+VI (λλλλadd= 3.0):MCARS+VI (∆∆∆∆g,add= var): Remark that FORM/SORM do not perfomr better in this example

238

U1

g 3

U2

1.66

Figure A.11.1: LSF g

Annex A.11: Concave failure domain

Limit state function:

(A.11.1)( ) ( )g u uu u

= − − −+

3 0 0 521 2

2 1 2. .

RandomVariables

mean(µ)

standarddeviation (σ)

PDF

u1u2

00

11

NN

Table A.11.1: Input Variables

Reliability method β # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)SORMSORM + ISMC (adsamp)MC CrudeMC+VI (σh=3.0)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

1.261.661.66*1.66*1.642.061.66*1.521.121.261.251.271.271.251.251.25

/87940718124941000010000011722602408555818

////////0.19/0.100.100.100.100.100.10

Table A.11.2: Results for different reliability methods (*:FORM/SORM in point (0,0) result in β=3.00)

Method ADI DARS (λadd= 3)

DARS(λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

1.311.841.71

1.271.661.803.00/

1.251.66ββββLSF = 1.660.38/full quadraticKKKK0

1.25/ββββLSF = 1.663.001.00full quadraticKKKK0

1.25/ββββLSF = 1.660.01.00full quadraticKKKK0

Table A.11.3: Intermediate results

MCARS+VI (λλλλadd= 3.0): After a limited number of LSFE, the estimated response surface is identical to the originallimit state function. Because λadd is equal to 3.00 and the failure function is very concave, most samples are in theimportant region in which a new limit state function evaluation is performed. This explains the high number of limitstate function evaluations. MCARS+VI (∆∆∆∆g,add= var): 18 samples to find the roots in the ADI procedure and to obtain a perfect fit of the LSF.Annex A.12: Series System with 4 branches

239

Limit state function: (A.12.1)

( ) ( )( ) ( )

( )( )

g

u u u u

u u u uu uu u

=

+ − − +

+ − + +− +− +�

min

. . /

. . /..

3 0 01 2

3 0 01 235 235 2

1 22

1 2

1 22

1 2

1 2

2 1

Random Variables mean (µ) standard deviation (σ) PDF

u1,u2 0 1 NTable A.12.1: Input Variables

Reliability method β # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)SORMSORM + ISMC (adsamp)MC CrudeMC+VI (σh=3)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

2.853.503.003.003.113.013.133.133.042.852.872.842.922.853.013.06

/32784948123131000010000016622717585224169

////////0.23/0.370.370.370.370.370.37

Table A.12.2: Results for different reliability methods

Method ADI DARS (λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

4.284.8114.5

2.923.005.033.00/

2.853.003.581.81/full quadratic0.35/3.88

3.01/4.313.002.42full quadratic0.42/3.88

3.06/4.012.22.32pure quadratic0.29/3.87

Table A.12.3: Intermediate results

DARS (λλλλadd= var): MCARS+VI (λλλλadd= 3.0) and MCARS+VI (∆∆∆∆g,add= var): because the LSF with its several branches is not describedaccurately with a second order polynomial, a relatively large number of LSFE is required. For more informationthe reader is referred to Chapter 3, or Annex A.14, which is similar.

240

Annex A.13: Parallel system

Limit state function: (A.13.1)g

u uu uu uu u

=

− −− −− −− −�

max

.

.

.

.

2 6772 5002 3232 250

1 2

2 3

3 4

4 5

Random Variables mean (µ) standard deviation (σ) PDF

u1, u2, u3, u4 0 1 NTable A.13.1: Input Variables

Reliability method β (pf) # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISSORMSORM + ISMC (adsamp)MC CrudeMC+VI (σh=3)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

3.52*1.85***3.73*3.523.553.583.523.523.27c (5.4 10-4)3.273.27

//1086///523/100001000002778581127314314314

////////0.05/0.490.490.490.490.490.49

Table A.13.2: Results for different reliability methods

Method ADI DARS (λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

Q

Q

3.25

3.523.493.863.00/

3.273.005.203.00/linear2.45/3.86

3.27/5.203.00σσσσh,0=1.71linear2.37/3.89

3.27/5.203.00σσσσh,0=1.71linear2.37/3.89

Table A.13.3: Intermediate results

DARS (λλλλadd= var), MCARS+VI (λλλλadd= 3.0) and MCARS+VI (∆∆∆∆g,add= var): The result is equal to that of theMCARS+VI method, because only the first two steps of the analysis are performed. There are no roots for theprincipal directions (see ADI in table A13.3). Therefore a full factorial design is performed (following the Matlabprocedure ff2n(n).m, (Matworks, 2000)). In this case there are 5 random variables. The full factorial designrequires 35 = 243 directions. So, after the principal directions are analysed and no roots found, another 35-5 = 238directions are studied. This requires substantially more effort then starting a random directional sampling asproposed by Waarts (Waarts, 2000), as can be seen from table A.13.2. Advantage is that each direction is takenevenly important. In case of directional sampling, an important direction might never be found afterwards.

241

Annex A.14: Altered series system with 4 branches - described in Chapter 2 and 3

Limit state function: (A.14.1)

( ) ( )( ) ( )

( )( )

g

u u u uu u u u

u uu u

=

+ − − ++ − + +− +− +�

min

. . /

. . /..

2 0 01 22 0 01 2

2 5 22 5 2

1 22

1 2

1 22

1 2

1 2

2 1

Random Variables mean (µ) standard deviation (σ) PDF

u1,u2 0 1 NTable A.14.1: Input Variables

Reliability method β (pf) # LSFE V(pf) σ(β)

ExactFORM FORM NLPQLFORM RFLSFORM RFLS + ISFORM-ARS (u-space)FORM-ARS (x-space)SORMSORM + ISMC (adsamp)MC CrudeMC+VI (σh=1.3)DSDARS (λadd= 3)DARS (λλλλadd= var)MCARS+VI (λλλλadd= 3.0)MCARS+VI (∆∆∆∆g,add= var)

1.702.02.02.02.02.02.02.142.141.61 (0.016)2.0 (0.064) (0.050)1.69 (0.045)1.91 (0.028)1.91 (0.028)1.84 (4.3 10-2)1.70 (4.5 10-2)

622983733515179192

0.150.150.150.150.150.150.150.15

Table A.14.2: Results for different reliability methods

Method ADI DARS (λadd= 3)

DARS (λλλλadd= var)

MCARS+VI(λλλλadd= 3.0)

MCARS+VI(∆∆∆∆g,add= var)

βλmin

βRS,FORM

λadd//∆g,add

σh,finRS-modelF/Fα

2.853.53.53.00

1.912.001.653.00/

1.652.002.051.48

full quadratic0.06/3.96

1.84/2.451.251.30full quadratic0.08/3.92

1.70/2.791.511.30full quadratic0.08/3.82

Table A.14.3: Intermediate results

DARS (λλλλadd= var); MCARS+VI (λλλλadd= 3.0); MCARS+VI (∆∆∆∆g,add= var): see Chapter 2 and Chapter 3

242

[1] [2]

[3]

P

Q

A1,I1,E

A2,I2,E

A1,I1,E

L= 6 m

H=4

m

ux δ

Annex A.15: Reliability of a plane frame DARS-CALFEM: a stochastic finite element method

A.15.1. Problem definitionThe reliability of a plane frame, subjected to a horizontal and vertical load, is analyzed using the DARS procedure.For the linear elastic analysis, the finite element code Calfem (Calfem, 2000) is used. This general purpose finiteelement code is implemented in MATLAB (Mathworks, 1999). The results are compared with the software resultsobtained with Powerframe (Buildsoft, 2001), that evaluates the structure using a limit states design conceptaccording to the European standards EC1 and EC3. A linear-elastic material behavior is assumed in all cases. Thestructure and its variables are outlined in figure A.15.1.

Figure A.15.1: Plane frame - geometry, boundary conditions and variables (adopted from [Calfem, 2000])

The random variables and their parameters are summarized in table A.15.1.

Type symbol[dimension]

description mean value Standard deviation cov[%]

PDF

A

R

S

A1 [m2]I1 [m4]A2 [m2]I2 [m4]E [N/m2]fy [N/m2]P [N]Q [N/m]

cross-sectional areamoment of inertiacross-sectional areamoment of inertiabending stiffnessyield stressHorizontal loadVertical load

45.3 10-4

2510 10-8

142.8 10-4

33090 10-8

210 109

235 109

500065000

0.453 10-4

50.2 10-8

1.428 10-4

661.8 10-8

4.2 109

4.7 109

4005200

12122288

NNNNNLNNN

Legend: A: Geometry; R: Resistance, S: LoadTable A.15.1: Random variables

A.15.2. Reliability analysisThe ultimate as well as the serviceability limit state are checked, according to EC3:

(A.15.1)

g f

gu H

L

ULS y y

SLS

x

= −

=−

−�

σ

δmin 150

200

243

The steel stresses (σy) are checked in 20 cross-sections evenly distributed over both columns [1] ,[2] and beam [3].

Because of the functional form of the bending stresses, the formulation of the ultimate limit state is altered. Thisis done to obtain a functional form that can be captured more easily using a 2nd order polynomial RS:

(A.15.2)( )σ σy ULS y yNA

MvI

g f= + = −log log( )

The results of the reliability analysis are summarized in Table A.15.2.

Results of DARS procedure Ultimate Limit State Serviceability Limit State

βσ(β)V(β)pfnLSFE

λADI,min

λmin

λadd,finF/FαRS-model

3.540.090.032.0 10-4

403.53= λADI,min 010-23

linear model

4.850.050.0086.3 10-7

325.074.780.1810-24

linear model

Table A.15.2: Reliability analysis using DARS - summary of results

In both cases a linear Response Surface is able to estimate the real behavior very accurately, λadd is small in bothcases, F/Fα as well. The number of limit state evaluations nLSFE remains limited for both analyses. Each limit stateanalysis requires a finite element calculation using Calfem. Because both programs (DARS and Calfem) areimplemented in the open Matlab program structure, the required exchange of information can be organized veryefficiently.

A.15.3. ConclusionsFor the ULS, following conclusions can be drawn:• The reliability index does not meet the target reliability index: β = 3.54 < βT = 3.70. This is confirmed

by the limit states analysis, according to EC3 using Powerframe. The material is exhausted for more then100 %, namely: 123 %. Disadvantage of the limit states design concept is that 123% percent does not tellanything about the resulting safety. The structure will not collapse. Its failure probability remains low:pf = 2.0 10-4.

• The most critical direction, as found in the ADI procedure (step 1 in the DARS procedure), results in theoverall minimum distance: λmin = 3.53. The failure point is reached when the distributed load (Q) equals:Q = 83.350 N/m.

For the SLS, following conclusions can be drawn:• The reliability index exceeds the target reliability index: β = 4.85 > βT = 2.1. This is confirmed by the

limit states analysis, according to EC3 using Powerframe. The maximum horizontal displacement ux

remains limited: ux = 2.7 mm = H/1485 < H/150. The maximum deflection remains limited: δ = 17 mm= L/353 < L/200.

• The most likely failure point does not coincide with one of the principal directions explored in ADI. Itis a combination of low bending stiffness (E) and high vertical distributed load (Q): [E,Q]=[200 109 N/m2,86974 N/m].

The combination of a finite element analysis and a reliability method results in a very powerful tool to assess thesafety of a structure. This is only possible when an efficient communication between both programs can beestablished. The open program structure of Matlab supplies a very powerful instrument for that purpose.

244

245

Figure B.1: Definition of E, fc and Gf,c

Annex B - Experimental results - summary andstatistical processingIn the following sections, the individual test results are summarized. The objectives are:- transparency,- provide the individual data from which the summary Tables in Chapter 5 are derived,- fill in the need of availability of data for research purposes.

The definitions of the Young’s modulus (E), the compressive strength (fc) and the fracture energy (Gfc) areillustrated in Figure B.1. Reference is made to a characteristic length h of 200 mm (Lourenço, 1996) to provide anobjective energy dissipation, overlapping with the reference length of the LVDT’s used for the masonry cores and

pillars. This allows mutual comparison of values from different sample heights. The values of the different samplesare re-scaled appropriate to the used reference length of deformation measurements used (LVDT or platedisplacement), according to the values listed in Table B.1.

Sample type sample height[mm]

measurement basis fordisplacement recording

factor for rescaling Gfc to areference height: href = 200 mm

brick-cores ‰40 44 h = 44mm 200/44=4.55

brick-couplets 120 h = 120 mm 200/120=1.67

brick-prisms 160 LVDT, lref = 100 mm 200/100=2

masonry-prisms 360 LVDT, lref = 200 mm 200/200=1

masonry-cores ‰150 mm 300 LVDT, lref = 200 mm 200/200=1

masonry-wallets 570 LVDT, lref = 460 mm 200/460=0.43Table B.1: re-scaling factor for Gfc

246

no. h [mm] φ [mm] mass [g] FV [kN] A [mm2] V [mm3] fc [N/mm2] ρ [kg/m3]

1 42.47 50.02 143.4 10.29 1965 83467 5.24 17182 40.58 50.20 144.9 10.16 1979 80301 5.13 18043 41.71 50.14 147.5 11.65 1974 82340 5.90 17914 41.68 49.96 143.3 8.79 1961 81719 4.48 17545 42.91 50.01 151.9 9.23 1964 84282 4.70 18026 44.03 50.11 157.3 8.41 1972 86839 4.26 18117 40.91 49.76 139.5 12.45 1944 79541 6.40 17548 41.02 50.02 146.9 13.51 1965 80607 6.88 18229 41.28 50.01 145.6 11.85 1964 81069 6.03 1796

10 41.32 49.95 146.8 11.2 1960 80969 5.72 181311 42.55 50.06 145.3 11.91 1968 83736 6.05 173512 42.84 50.13 150.2 13.42 1973 84543 6.80 177713 49.16 50.08 152.3 11.61 1970 96828 5.89 157314 42.35 50.03 141.4 10.4 1966 83243 5.29 169915 41.99 49.84 143.6 9.4 1951 81920 4.82 175316 41.30 50.03 148.1 15.99 1966 81190 8.13 182417 41.12 50.11 142.7 12.8 1972 81105 6.49 175918 41.36 50.20 144.3 11.75 1979 81856 5.94 176319 41.03 50.09 145.4 12.7 1971 80858 6.44 179820 41.12 50.05 144.8 8.97 1967 80900 4.56 179021 40.18 50.07 144.2 17.34 1969 79104 8.81 182322 41.99 50.08 150.4 14.73 1970 82706 7.48 181823 41.10 50.13 145.6 16.79 1974 81120 8.51 179524 41.31 49.74 140.3 13.6 1943 80265 7.00 174825 41.61 49.83 147.3 17.39 1950 81157 8.92 181526 41.76 49.98 147.9 20.7 1962 81930 10.55 180527 38.21 49.96 132.1 12.39 1960 74895 6.32 176428 36.82 49.96 129.2 11.86 1960 72166 6.05 179029 40.40 49.99 145.4 14.87 1963 79288 7.58 183430 40.22 50.07 146.3 15.83 1969 79193 8.04 184731 41.71 49.90 142.2 14.69 1955 81559 7.51 174432 40.96 50.26 142.8 35.37 1984 81263 17.83 175733 41.45 49.82 144.6 15.22 1950 80807 7.81 178934 42.03 50.32 147.2 23.84 1989 83580 11.99 176135 40.65 50.24 144.6 12.63 1982 80584 6.37 179436 40.50 50.07 143.9 10.75 1969 79755 5.46 180437 40.96 50.07 141.2 9.78 1969 80661 4.97 175138 43.01 50.12 150.2 12.84 1973 84861 6.51 177039 41.60 50.00 140.8 9.37 1963 81676 4.77 172440 42.91 50.23 148.8 11.74 1982 85036 5.92 175041 44.02 50.13 149 7.8 1973 86872 3.95 171542 43.27 50.10 144 6.27 1971 85284 3.18 168843 43.12 50.21 144 8.54 1980 85362 4.31 168744 43.22 50.11 149 11.31 1972 85247 5.73 174845 41.61 50.14 144.5 9.44 1975 82165 4.78 175946 42.78 50.17 142.8 7.26 1977 84559 3.67 168947 45.01 50.11 149 12.04 1972 88754 6.11 167948 43.56 50.21 142 8.5 1980 86250 4.29 164649 43.53 50.19 151.2 9.05 1979 86133 4.57 175550 42.70 50.15 140.1 10.09 1976 84356 5.11 166151 43.51 50.25 142 8.56 1983 86288 4.32 1646

x 41.93 50.07 145.05 12.49 1969 82553 6.34 1758.71s 1.74 0.13 4.63 4.69 10 3571 2.37 56.07cov 4 0 3 38 1 4 37 3Table B.2: Cores ‰50 with height 44 mm - test results and summary

Figure B.2: Cores ‰50

B.1. Cores ‰50 with a height of 44mm

Histogram, Normal probability plot, Lognormal probability plot: Figure 5.3

UH-plot, Figure 5.4Pareto QQ-plot, Figure 5.5Hill-estimation, Figure 5.6

247

0

12

3

4

56

7

8

9

10

-0.01 0 0.01 0.02 0.03 0.04 0.05Strain ε [mm/mm]

σ X

Y

Figure B.3: Couplets - stress-strain relationship and definition of material parameters

Figure B.4: Brick geometry - histogram and estimated PDF-plot

B.2. Couplets with a height of 120 mm

brick geometry length width height

sample mean: [mm]xsample spread: s [mm]coefficient of variation: cov [%]number of test samples: nestimated PDF

189.172.081.1400N

86.281.902.2400N

44.211.252.8400N

Table B.3: Brick geometry - statistical summary of test results

248

no. fc [MPa] Ey [MPa] Gfc[N/mm]1 5.30 417 1.552 4.59 391 1.433 8.28 603 1.264 6.63 399 2.155 3.43 266 3.096 9.07 630 3.267 4.30 341 1.768 4.40 327 3.079 6.49 428 1.64

10 7.30 496 4.8711 4.34 364 1.6712 2.96 281 1.6513 5.63 326 2.2714 4.80 357 1.7815 7.06 285 2.9316 5.21 422 2.0217 8.28 564 5.6318 4.12 285 3.5319 3.90 279 1.9220 5.87 507 2.5221 5.73 330 2.8022 3.88 317 2.1323 8.27 542 4.8824 3.68 267 1.4725 2.87 212 1.2926 4.58 473 2.0927 6.09 499 2.2828 4.88 337 1.9629 4.74 369 2.2430 5.21 311 2.4831 3.16 267 1.7632 7.06 512 2.7633 5.91 351 1.4934 6.02 470 1.4835 3.42 324 2.1936 4.17 343 1.8637 3.07 167 1.6438 3.88 378 1.5839 4.56 417 2.0440 5.19 461 2.2541 4.29 350 2.3442 4.91 355 2.0343 3.83 271 1.5444 2.82 165 1.6845 4.70 339 2.5346 4.37 287 2.6647 7.94 546 2.6548 4.66 399 0.5949 5.56 511 3.5150 6.46 541 2.03

x 5 382 2s 2 109 1cov 30 29 41ρ fc [MPa] Ey [MPa] Gfc[N/mm]fc 1Ey 0.83 1Gfc 0.51 0.37 1

Table B.4: Couplets test results

Figure B.5: Histogram, Normal and Lognormal probability plots - fc

Figure B.6: Histogram, Normal and Lognormal probability plots - E

Figure B.7:Histogram, Normal and Lognormal probability plots - Gfc

249

no. h [mm] lref [mm] w [mm] d [mm] FV [kN] fc,x [MPa] Ex [MPa]1 160.43 115.19 45.22 21.57 5.96 6.11 15432 160.28 113.25 45.84 23.37 8.46 7.90 15403 159.16 108.69 45.42 23.72 8.68 8.064 160.36 115.40 43.62 25.96 8.96 7.91 16105 160.26 114.27 46.92 25.58 14.37 11.97 27636 160.1 114.21 44.09 27.27 7.95 6.61 17507 160.41 114.28 45.67 25.59 10.83 9.27 18278 158.68 115.50 45.70 26.02 10.05 8.45 18689 159.15 115.26 44.36 29.65 9.65 7.34 1748

10 158.92 113.65 43.56 24.90 7.26 6.69 155211 159.87 115.36 44.69 25.46 12.16 10.69 240712 157.82 114.19 45.46 25.98 10.69 9.05 176013 159.28 114.80 45.19 24.28 8.78 8.00 196314 158.77 115.92 44.80 29.08 11.83 9.08 180215 160.33 113.58 44.77 23.30 9.02 8.65 172316 157.65 115.20 45.23 26.10 1.92 1.63 17 160.06 113.99 45.70 24.47 9.32 8.34 181218 160.73 115.87 45.24 28.09 9.33 7.34 133519 160.48 115.76 44.88 24.43 10.10 9.21 191220 159.68 116.80 45.50 25.12 7.05 6.17 154521 160.45 113.51 45.39 27.00 9.21 7.51 136722 158.44 115.88 45.53 27.65 5.66 4.50 87023 159.42 114.63 45.68 27.00 10.16 8.24 175324 160.44 113.75 44.55 28.95 17.33 13.44 274025 160.29 114.88 45.51 29.22 10.76 8.09 148326 158.62 115.46 45.12 24.44 9.45 8.57 125927 160.05 117.01 46.14 23.89 7.54 6.84 150528 159.94 114.98 44.07 26.67 11.51 9.80 251429 158.91 114.22 46.26 25.66 10.61 8.94 230830 158.88 116.32 44.92 24.41 8.53 7.78 146731 158.25 114.98 45.26 25.40 8.52 7.41 148232 160.01 115.01 45.04 24.36 10.59 9.65 182733 159.17 115.17 45.04 25.56 8.32 7.23 124134 159.08 114.88 45.48 28.57 12.40 9.55 179935 159.27 114.97 45.37 25.01 7.80 6.87 156336 160.23 115.23 43.69 28.10 10.13 8.25 157537 160.09 114.98 44.48 24.82 5.66 5.13 138538 160.11 113.27 45.60 28.30 11.20 8.68 149739 160.52 114.86 46.08 26.72 8.91 7.24 135340 160.49 114.34 44.65 24.15 8.24 7.64 1596

x 159.63 45.14 25.89 9.37 8.00 1712s 0.82 0.72 1.88 2.53 1.93 399cov 0.5 2 7 27 24 23Table B.5: Brick prisms h=160 mm - test results and summary

B.3. Prisms sawn from bricks with a height of 160 mm

250

Figure B.8: Histogram, Normal and Lognormal probability plot - fc,x

Figure B.9: Histogram, Normal and Lognormal probability plot - Ex

251

no. ft [N/mm2] fc,left [Mpa] fc,right [Mpa] Eleft [Mpa] Eright [Mpa] Ec,left [Mpa] Ec,right [Mpa]1 3.12 8.13 7.76 373 435 826 12092 2.78 9.58 9.59 360 377 756 8363 2.59 8.60 8.19 338 452 678 13704 2.40 6.21 5.91 401 448 1019 13915 2.29 5.91 6.22 194 387 278 9836 1.93 6.51 6.73 427 403 1325 11217 2.84 7.83 7.85 483 408 1996 11348 2.28 8.04 8.56 408 422 1134 12509 2.12 6.96 8.46 484 484 2088 2088

10 2.94 8.05 8.10 525 490 2966 211411 3.17 7.83 7.54 542 330 3062 66112 2.90 7.18 7.69 312 62513 2.92 7.59 7.88 365 521 860 294014 2.76 7.51 7.54 334 408 701 113315 2.87 8.45 8.48 530 412 2991 114516 2.65 7.31 7.69 300 443 556 137517 8.46 7.35 400 383 1111 98918 2.19 7.89 7.89 432 408 1342 113519 2.12 6.68 6.89 310 354 596 78520 2.36 6.39 6.37 292 253 525 41121 2.60 6.77 5.94 328 459 557 108322 2.03 5.27 7.53 238 369 379 87123 2.43 8.27 7.26 496 225 2138 34624 2.32 6.79 8.16 280 503 488 216925 2.12 8.63 7.70 444 414 1412 115126 1.96 7.13 6.99 408 408 1134 113427 2.34 6.84 7.38 446 446 1419 141928 2.37 7.66 6.86 488 309 2017 59529 2.40 6.63 7.41 308 370 593 87330 2.29 7.11 7.32 426 389 1289 100631 1.56 4.99 5.30 414 311 1151 59832 1.99 4.86 5.93 263 375 446 91033 2.09 5.39 5.87 368 409 868 113534 2.06 7.46 6.44 390 366 1006 86435 2.06 7.14 6.59 406 332 1128 69836 2.28 7.63 7.41 379 357 948 82037 2.21 8.36 9.23 363 363 857 85738 2.60 8.71 8.97 346 458 758 163739 2.66 9.84 9.56 475 463 1816 165440 3.47 11.47 11.55 449 449 1604 160441 3.54 10.83 10.84 431 421 1373 127642 2.99 10.40 10.73 467 373 1666 88043 3.51 10.66 11.62 329 362 691 85444 3.63 10.86 11.38 404 519 1122 293245 3.26 10.23 10.80 445 445 1591 159146 2.95 9.96 10.26 448 512 1599 288947 2.68 9.45 9.36 439 439 1568 156848 3.00 9.67 10.18 404 424 1284 151449 2.53 11.19 12.45 454 567 1621 567350 3.36 10.96 9.99 439 423 1475 131351 3.69 11.04 12.33 536 536 3024 302452 2.42 8.82 11.33 481 571 1987 571153 2.37 10.36 10.48 492 455 2220 162554 2.73 10.69 10.88 453 518 1618 2923

x 2.60 8.31 410 1398s 0.49 2 75 898cov 19.0 22 18 64Table B.6: Mortar beams 1st batch - test results and statistical summary

B.4. Mortar samples - l×w×h = 160×40×40 mm3

252

Figure B.10: Mortar - Histogram, Normal and Lognormal Probability plot - ft

Figure B.11: Mortar - Histogram, Normal and Lognormal Probability plot - fc,y

253

no. fc,x [MPa] fc,y,left [MPa] fc,y,right [MPa] E [MPa]1 3.10 4.49 20192 3.80 5.43 4.80 26743 3.73 5.48 4944 4.63 5.95 10065 3.62 4.00 4.23 19486 4.97 5.66 6.55 15987 4.66 5.01 5.44 12988 4.43 5.86 4.91 23289 3.98 4.81 4.88 4184

10 4.64 5.51 5.21 128411 4.24 5.24 5.76 62612 4.14 5.74 5.71 165913 3.72 4.08 3.89 95014 3.52 4.85 178415 4.38 5.93 5.31 114216 3.42 4.29 3.47 105317 3.99 4.90 4.87 189018 3.43 4.64 3.97 71719 3.22 4.39 3.90 62920 2.85 4.84 1277

x 3.92 4.97 1388s 0.58 0.72 611cov 15 15 44Table B.7: mortar samples, 2nd batch - test results

Figure B.12: Mortar, Histogram, Normal and Lognormal probability plot - E

254

no. fc,y [MPa] Ey,LVDT [MPa] Ey,plate [MPa] Gfc [N/mm]1 4.09 1735 909 1.552 5.00 1690 974 1.753 3.90 1295 820 0.874 5.33 1629 1020 1.805 2.91 1954 1143 1.016 3.77 1555 571 1.447 4.82 1593 877 1.638 2.86 817 500 1.249 4.69 1984 806 2.40

10 3.25 1234 651 0.8511 4.26 1548 870 2.6012 3.48 1208 717 1.3713 5.47 2387 1070 2.1814 3.79 1355 794 1.3315 5.75 3009 1154 1.9916 4.24 1643 867 1.8417 4.92 2119 743 2.7418 4.30 1368 673 2.1819 4.11 750

x 4.26 1673.44 837.32 1.71s 0.83 496.55 180.92 0.56cov 19 30 22 33Table B.8:Masonry pillars - test results and statistical summary

Figure B.13: Pillars, Histogram, Normal and Lognormal probability plot - fc

B.5. Masonry pillars

255

Figure B.14: Pillars, Histogram, Normal and Lognormal probability plot - Ey,LVDT

Figure B.15: Pillars, Histogram, Normal and Lognormal probability plot - Gfc

256

no. fc,y [Mpa] Ey,LVDT [Mpa] Ey,plate [Mpa] Gfc [N/mm]

1 5.6284 2053 1065 3.072 4.5312 2729 842 1.403 3.5838 1146 574 0.784 4.8181 1182 773 0.995 4.1149 1339 735 0.49

x 4.54 1690 798 1.35s 0.77 687 179 1.02cov 17 41 22 76Table B.9: Masonry cores - test results and statisticalsummary

Figure B.16: Crack pattern wallet V1

Figure B.17: Crack pattern wallet V2

B.6. Masonry cores ‰‰‰‰=150 mm - vertically drilled

B.7. Masonry wallets - uni-axial compressive test

Wallets V1 V2 V3

height [mm]width [mm]thickness [mm]Maximum load FV,max [kN]

569575185617

572578186609

569575185716

Table B.10: Masonry wallets V1-V3 - geometry and Vertical loading

Test results and statistical summary, Table 5.13.

257

no. τmax G [MPa] Gf,II [N/mm]

1 2.51 841 0.552 2.01 776 1.43 2.16 854 0.194 2.36 1335 0.675 3.13 609 1.52

x 2.43 883.00 0.87s 0.4 270.8 0.6cov 18 31 66Table B.11: Masonry cores - testresults and statistical summary

Figure B.18: Crack pattern wallet V3

B.8. Masonry cores ‰‰‰‰=150 mm - diagonally drilled

B.9. Masonry wallets - shear test

Wallets D1 D2 D3

height [mm]width [mm]thickness [mm]First crack appearance FV,c [kN]Maximum load FV,max [kN]

575571184107162.7

575571184175187.3

578577181135146.7

Table B.12: Masonry wallets D1-D3- geometry and vertical loading

Test results and statistical summary, Table 5.13.

258

Figure B.19: Diagonal wallet D1 - crack pattern

Figure B.21: Diagonal Wallet D3 - crack pattern

Figure B.20: Diagonal Wallet D2 - crack pattern

259

no. φ [mm] ft [N/mm2] failure1 49.71 0.26 b2 49.75 0.21 a3 49.39 0.26 b4 49.77 0.36 a5 49.66 0.26 b6 49.92 0.41 d7 49.45 0.42 b8 49.65 0.52 a9 49.65 0.31 a

10 49.62 0.05 b11 49.62 0.16 a12 49.78 0.10 a13 49.63 0.10 b14 49.47 0.16 b15 49.54 0.10 b16 49.93 0.41 a17 49.65 0.26 a18 49.65 0.36 a19 49.65 0.31 a20 49.65 0.26 a21 49.65 0.21 a22 49.61 0.26 d23 49.64 0.21 a24 49.53 0.21 d25 49.65 0.36 a26 49.32 0.31 a27 49.72 0.36 a28 49.50 0.47 a

29 49.86 0.20 a30 49.66 0.36 a31 49.75 0.21 a32 49.83 0.26 a33 49.48 0.16 b34 49.65 0.31 a35 49.65 0.26 a36 49.51 0.47 d37 49.65 0.31 a38 49.65 0.41 a39 49.65 0.41 a40 49.83 0.31 a41 49.71 0.26 a42 49.65 0.26 a43 49.65 0.31 a44 49.65 0.26 d45 49.36 0.37 d46 49.65 0.36 a47 49.52 0.42 a48 49.83 0.15 b49 49.39 0.26 c/d50 49.81 0.26 a51 49.65 0.31 d52 49.53 0.26 a53 49.81 0.26 a54 49.71 0.21 a55 49.78 0.31 a56 49.63 0.26 a

x 49.65 0.28s 0.13 0.10cov 0.27 35

B.10. Masonry cores - tension

Table B.13: Direct tensile tests - testresults and statistical summary Legend:(a) failure in the brick, (b) failure in the mortar, (c+d) failure in the brick mortar interface

260

Figure B.22: Cores, Histogram, Normal and Lognormal probability plot - ft

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no. σ3/σ1 σ1,max σ3,max Remarks Sample name coating1 0.00 5.98 0.00 GF moi03a rubber2 0.00 5.03 0.00 GF moi10a rubber3 0.00 4.98 0.00 GF moiii01a rubber4 0.00 5.03 0.00 GF moiii04a rubber5 0.05 8.93 0.47 GF moii15a rubber6 0.10 11.48 1.16 GF moi04a rubber7 0.10 12.89 1.35 GF moii13a rubber8 0.11 9.25 1.00 GF moii05a rubber9 0.11 12.20 1.36 GF moii06a rubber

10 0.11 9.92 1.14 GF moiii02a rubber11 0.12 11.70 1.35 GF moi05a rubber12 0.12 11.51 1.36 GF moiii03a rubber13 0.15 13.69 2.05 GF moii07a rubber14 0.16 13.14 2.05 GF moi06a rubber15 0.18 14.41 2.59 GF moii14a rubber16 0.19 12.36 2.36 GF moiii04a rubber17 0.25 16.43 4.14 GF moi07a rubber18 0.25 16.23 4.09 GF moi12a rubber19 0.25 15.65 3.95 GF moiii05a rubber20 0.25 17.38 4.40 GF moii08a rubber21 0.49 18.42 9.07 GF moi08a rubber22 0.49 19.92 9.83 GF moii09a rubber23 0.50 20.60 10.20 GF moiii06a rubber24 0.50 20.37 10.13 σ3max moi11a rubber25 0.73 15.67 11.45 σ3max moii10a rubber26 0.74 17.32 12.79 σ3max moi09a rubber27 0.98 12.14 11.86 σ3max moii11a rubber28 0.98 12.24 12.02 σ3max moii12a rubber

Table B.14: Triaxial tests on masonry cores - test resultsLegend: GF: global failure; σ3max: maximum confining pressure reached;Bold sample names: σ-ε relationship used for Figure 5.27

B.11. Triaxial testing - mortar cores

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no. σσσσ3/σσσσ1 σσσσ1,max σσσσ3,max End of test Sample name Coating1 0.00 5.63 0.00 GF corev1 none2 0.00 4.53 0.00 GF corev2 none3 0.00 3.58 0.00 GF corev3 none4 0.00 4.81 0.00 GF corev4 none5 0.00 4.11 0.00 GF corev5 none6 0.26 9.62 2.47 GF corev2b rubber7 0.26 8.94 2.34 GF corev5b rubber8 0.25 15.57 3.88 GF corev27a lead and rubber9 0.40 14.05 5.57 LF corev10b textile and rubber

10 0.40 12.99 5.21 GF corev13b textile and rubber11 0.40 14.00 5.58 LF corev21a textile and rubber12 0.50 20.23 10.12 GF corev28d textile and rubber13 0.52 6.88 3.55 GF corev15a rubber14 0.49 7.88 3.90 GF corev17a rubber15 0.60 11.32 6.84 LF corev6e textile and rubber16 0.60 15.34 9.19 GF corev7d textile and rubber17 0.60 14.87 8.98 LF corev9c textile and rubber18 0.79 7.56 5.94 GF corev18c textile and rubber19 0.74 11.42 8.42 GF corev29c textile and rubber20 0.79 10.80 8.58 GF corev8a rubber21 1.00 12.59 12.58 σ3max corev12e Al and rubber22 1.00 12.61 12.59 σ3max corev30b Al and rubber23 1.01 12.27 12.37 σ3max corev31b Al and rubber24 1.99 7.36 14.62 σ3max corev12d Al and rubber25 2.00 1.48 2.96 LF corev10a rubber26 1.82 2.16 3.94 LF corev13a rubber27 3.94 3.80 14.97 σ3max corev12c Al and rubber28 3.93 2.67 10.48 LF corev19b textile and rubber29 12.48 0.65 8.11 LF corev16c textile and rubber30 23.14 0.65 15.04 σ3max corev20b textile and rubber

Table B.15: Triaxial tests on vertically drilled masonry cores - test results, Legend: GF: global failure; LF: local failure; σ3max: maximum confining pressurereached; sample name: final character assigns the number of tests performed on thesample

B.12. Triaxial testing - vertically drilled masonry cores

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no. σσσσ3/σσσσ1 σσσσ1 σσσσ3 Test end Sample name Coating1 0.00 5.03 0.00 GF cored1 none2 0.00 4.01 0.00 GF cored2 none3 0.00 4.32 0.00 GF cored3 none4 0.00 4.72 0.00 GF cored4 none5 0.00 6.26 0.00 GF cored5 none6 0.25 9.08 2.28 GF cored1a rubber7 0.25 8.03 2.00 GF cored5b rubber9 0.25 9.27 2.31 GF cored2a rubber8 0.50 14.29 7.11 GF cored3b textile and rubber

11 0.50 6.69 3.33 GF cored14a rubber12 0.50 11.28 5.63 GF cored15a textile and rubber10 0.75 17.70 13.24 σ3max cored11b textile and rubber13 0.75 17.75 13.27 GF cored18c textile and rubber14 0.75 15.29 11.46 σ3max cored8a textile and rubber15 1.00 12.33 12.33 σ3max cored9a textile and rubber19 1.00 12.38 12.38 σ3max cored19b textile and rubber20 1.00 12.45 12.45 σ3max cored20a textile and rubber16 3.93 3.83 15.04 σ3max cored12a textile and rubber17 14.29 0.63 9.00 σ3max cored13a textile and rubber18 18.68 0.63 11.77 σ3max cored20a textile and rubber

Table B.16: Triaxial tests on diagonally drilled masonry cores - test resultsLegend: GF: global failure; LF: local failure; σ3max: maximum confining pressurereached; sample name: final character assigns the number of tests performed on thesample

B.13. Triaxial testing - diagonally drilled masonry cores

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Curriculum vitae

Personals

Surname Schueremans Given name LucPlace of birth Merksem, BelgiumDate of brith May 12, 1970Married to Pascale InnocentChildren Bas (18/01/1999), Judith (31/07/2000)office address Department of Civil Engineering

Kasteelpark Arenberg 40, 3001 HeverleeTel: +32 16 32 16 79 Fax: +32 13 32 19 76e-mail: luc.schueremans@bwk.kuleuven.ac.be

Education

High school 1982-1988: Humaniora, Latijnse-Griekse, Sint-Michielscollege, Brasschaat1988-1989: Voorbereidend jaar wiskunde,Sint-Jan Bergmanscollege,Antwerpen

University 1989-1995: Katholieke Universiteit Leuven, Civil Engineering (BurgerlijkBouwkundig Ingenieur, optie Gebouwentechniek), Great DistinctionThesis: Systeemidentificatie van constructies (promotor: Prof. dr. ir. G. De Roeck)1992-1995: Aggregatie voor het onderwijs, Distinction

Career

Jobtitle 1995-1997: scientific collaborator (Wetenschappelijk medewerker) LRD (LeuvenResearch & Development).1998-2001: research assistant - Research grant offered by the Institute for theencouragement of Innovation by Science and Technology in Vlaanderen (IWT-Vlaanderen: Instituut voor de aanmoediging van Innovatie door Wetenschap enTechnologie in Vlaanderen)Ph. D. research entitled: “Probabilistic evaluation of structural unreinforcedmasonry” (probabilistische evaluatie van structureel ongewapend metselwerk)1998 May-July: visiting researcher at the University of Calgary, Canada1999 November-2000 July: part time visiting researcher at TNO Delft,Netherlands

Research Reliability analysisbuilding technology for structural restoration

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List of Publications

International journals

1. Schueremans L., Van Gemert D. and Van Dyck J., “A probabilistic evaluation method forstructural masonry”, International Journal for Restoration of Buildings and Monuments,Internationale Zeitschrift für Bauinstandsetzen, Aedificatio Verlag - Fraunhofer IRB Verlag, Vol.3, No. 6, pp. 553-567, 1997.

2. Van Gemert D., Toumbakari E. and Schueremans L., “Konstruktive Injektion von historischemMauerwerk mit mineralisch- oder polymergebundenen Mörteln”, International Journal forRestoration of Buildings and Monuments, Internationale Zeitschrift für Bauinstandsetzen,Aedificatio Verlag - Fraunhofer IRB Verlag, Vol. 5, No. 1, pp. 73-98, 1999.

3. Schueremans L., Van Gemert D. and Smars P., “Safety assessment of masonry arches usingprobabilistic methods”, International Journal for Restoration of Buildings and Monuments,Internationale Zeitschrift für Bauinstandsetzen, Aedificatio Verlag - Fraunhofer IRB Verlag,2001, submitted and reviewed.

4. Schueremans L., Van Rickstal F., Van Gemert D. and Venderickx K., “Evaluation of masonryconsolidation by geo-electrical difference mapping”, RILEM, Materials and Structures, 2001,submitted.

Book contributions

1. Van Gemert D. and Schueremans L., “Water repellent impregnation as chloride barrier inmarine concrete structures: field data and lifetime prediction”, Liber Amicorum, Prof. Sasse,March, 1999.

2. Schueremans L. and Van Gemert D., “Structureel gedrag. Metselwerk”, HandboekOnderhoud, Renovatie en Restauratie, Hoofdstuk II, II.2, pp. 1-50, Kluwer Editorial, december2000.

3. Schueremans L., Ignoul S. and Van Gemert D., “Structureel gedrag. Hout”, HandboekOnderhoud, Renovatie en Restauratie, Hoofdstuk II, II.2., pp.1-48, Kluwer Editorial, 2001.

4. Ignoul S., Schueremans L. and Van Gemert D., “Houtherstelling”, Handboek Onderhoud,Renovatie en Restauratie, Hoofdstuk II, Kluwer Editorial, in press, 2001.

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International conference proceedings

1. Peeters B., De Roeck G., Pollet T. and Schueremans L., “Stochastic Subspace Techniquesapplied to parameter identification of civil engineering structures”, Proceedings of “NewAdvances in modal synthesis of large structures: non-linear, damped and non-deterministiccases”, Lyon, pp. 151-162, 5-6 October, 1995.

2. Schueremans L. and Van Gemert D., “A probabilistic model for reliability evaluation ofhistorical masonry”, Proceedings of the 4th International Colloquium: Materials Science andRestoration, Esslingen, 17-19 December, 1996.

3. Schueremans L. and Van Gemert D., “Protection of new container terminal at Zeebruggeagainst Chloride ingress”, Structural Faults + Repair - 97, 7th International Conference andExhibition, Edinburgh, Scotland, Vol. 2, pp. 221-231, 8-10 July, 1997.

4. Schueremans L. and Van Gemert D., “Service Life Prediction Model for Reinforced ConcreteConstructions, Treated with Water-repellent Compounds”, ConChem Conference and Exhibition,Düsseldorf, Germany, 2-4 December, 1997.

5. Schueremans L. and Van Gemert D., “Reliability Analysis in Structural MasonryEngineering”,IABSE Colloquium “Saving Buildings in Central and Eastern Europe”, Berlin, July1998.

6. Van Gemert D. and Schueremans L., “Evaluation of the Water Repellent Treatment, Appliedas Chloride Barrier on a Quay-Wall at Zeebrugge Harbor”, Water Repellent Treatment ofBuilding Materials, Proceedings of Hydrophobe II, Zurich, Aedificatio Publishers, pp.91-106,September, 1998.

7. Van Gemert D., Toumbakari E. and Schueremans L., “Strukturelle Injektion von HistorischemMauerwerk mit mineralischen oder polymerischen Mörteln”, Bauhaus - Universität Weimar,Vorlesung “Grundlagen der Bauwerkserhaltung”, Weimar, 7 July, 1998.

8. Van Gemert D., Herroelen B., Schueremans L. and Dereymaeker J., “Strengthening of woodenstructural elements by means of polymer concrete”, ICPIC Conference, Bologna, September 14-18, pp. 577-588, 1998.

9. Schueremans L., Van Gemert D. and Beeldens A., “Accelerated chloride penetration test asa basis for service life prediction model for R/C constructions”, Sheffield, pp.1269-1280, June28-July 2, 1999.

10. Schueremans L., Van Gemert D. and Maes M.A., “Evaluating the reliability of structuralmasonry elements using response surface techniques”, 8DBMC, 8th International Conference onDurability of Building Materials and Components, May 30 - June 3, Vancouver, Canada,pp.1330-1342, 1999.

11. Schueremans L. and Van Gemert D., “Service Life Prediction of Reinforced concretestructures, based on in-service chloride penetration profiles”, 8DBMC, 8th International

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Conference on Durability of Building Materials and Components, may 30 - June 3, Vancouver,Canada, pp. 83-93, 1999.

12. Maes M.A., Schueremans L. and Van Balen K., “Reliability based assessment of an existingmasonry structure using performance limit states”, ICASP8, 8th International Conference onApplications of Statistics and Probability, Sydney, pp. 689-695, 12-15 December, 1999.

13. Van Gemert D., Schueremans L., Hitze K. and Standke B., “Hydrophobic treatment aschloride penetration barrier”, Structural Faults and Repair ‘99, 13-15 July, Kensington, London,England, Cdrom, 1999.

14. Schueremans L., Van Gemert D. and Huyse L.J., “Homogenization technique forunreinforced masonry - a stochastic extension”, Valenciennes, France, Numerical modeling ofuncertainties, November, 1999.

15. Schueremans L., Van Balen K., Hayen R., Peeters V. and Wittouck R., “The ‘sHertogenmolens water mills, monitoring of a historical monument for assessment of structuralsafety”, 5th Internationales Colloquium, Materials Science and Restoration - MSR ‘99, December1999.

16. Schueremans L., Hayen R., Van Gemert D. and Van Balen K., “Triaxial testing of masonryand lime mortars, test setup and first results”, 5th Internationales Colloquium, Materials Scienceand Restoration - MSR ‘99, December 1999.

17. Schueremans L., Van Gemert D., Soons B., Kustermans B. and Leeuwerck D., “Lasercleaning as a tool in conservation”, 5th Internationales Colloquium, Materials Science andRestoration - MSR ‘99, December 1999.

18. Schueremans L. and Van Gemert D., “Effectiveness of hydrophobic treatment as chloridepenetration barrier - on site investigations and service life prediction model”, 5th InternationalesColloquium, Materials Science and Restoration - MSR ‘99, December 1999.

19. Schueremans L. and Van Gemert D., “Effectiveness of hydrophobic treatment as chloridepenetration barrier - Comparison of laboratory results and in site investigations on a quay-wallin marine environment”, Materials Week, International Congress on Advanced Materials, theirProcesses and Applications, Munich, 25-29 September 2000.

20. Schueremans L. and Van Gemert D., “Assessment of existing masonry structures usingprobabilistic methods - state of the art”, STRUMAS V, The Fifth International Symposium onComputer Methods in Structural Masonry, 18-20 April, 2001.

21. Schueremans L. and Van Gemert D., “Assessment of existing masonry structures usingprobabilistic methods - new approaches”, STRUMAS V, The Fifth International Symposium onComputer Methods in Structural Masonry, 18-20 April, 2001.

22. Schueremans L. and Van Gemert D., “Predicting masonry properties from componentproperties using probabilistic techniques”, STRUMAS V, The Fifth International Symposium

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on Computer Methods in Structural Masonry, 18-20 April, 2001.

23. Hayen R., Schueremans L., Van Balen K. and Van Gemert D., “Triaxial behavior of historicmasonry”, STRUMAS V, The Fifth International Symposium on Computer Methods inStructural Masonry, 18-20 April, 2001.

24. Schueremans L., Smars P. and Van Gemert D., “Safety of arches - a probabilistic approach”,9CMS, 9th Canadian Masonry Conference, New Brunswick, Canada, June, 2001.

25. Van Gemert D., Van Rickstal F., Venderickx K. and Schueremans L., “Geo-electricalmeasurements as non-destructive test method”, WTA, Zurich, April, 2001.

26. Hayen R., Schueremans L., Van Balen K. and Van Gemert D., “Triaxial testing of historicmasonry, test set-up and first results”, STREMAH V, Structural Studies, Repairs andMaintenance of Historical Buildings, 28-30 May, Bologna, 2001.

27. Van Gemert D., Van Rickstal F., Schueremans L. and Toumbakari E., “Non-destructiveevaluation and consolidation of natural stone masonry”, Proceedings Expertengespräch zumThema “Fugen - nachhaltige Fugeninstandsetzung mit innovativen Technologien für dasVölkerschlachtdenkmal”, Leipzig, 20 p., 16th February, 2001.

28. Schueremans L. and Van Gemert D., “Reliability based evaluation and desing - SFEM usingMatlab”, Nordic Matlab Conference, October 17-18, Oslo, pp. II.217-II.221, 2001.

National journals

1. Van Gemert D. and Schueremans L., “Preventieve bescherming van gewapend betonstructurentegen chloriden-indringing: Evaluatie op container-terminal Zeebrugge”, Infrastructuur in hetLeefmilieu, Ministerie van de Vlaamse Gemeenschap, 2/97, pp.93-100, 1997.

2. Schaerlaekens S., De Bruyn R., Schueremans L. and Van Gemert D., “Beoordelen van oudmetselwerk: onderzoek in het kader van het programma ‘Restauratie van Buitenmuren’”, WTCB-Tijdschrift, 99/2, 15 blz 3-13, 1999.

3. Van Gemert D., Schueremans L. and Toumbakari E., “Structurele injecties van waardevolmetselwerk met minerale- of kunststofgrouts”, ICOMOS CONTACT, driemaandelijksinformatieblad van Icomos Vlaanderen/Brussel vzw, jaargang 11, nr. 2/1998, 16 p., 1998.

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National conferences

1. Van Gemert D. and Schueremans L., “Structurele injecties van waardevol metselwerk metminerale- of kunststofgrouts”, KVIV studiedag, “Restauratie, historische materialen enhedendaagse technieken”, 26 November, 1997.

Tutorial notes

1. Schueremans L., “De response surface methode als een basis voor de beoordeling van debetrouwbaarheid van structuren - theorie en toepassingen”, Postacademisch Onderwijs,Probabilistisch ontwerpen, TUDelft, 7,8,14,15 September, case 10, pg. 1-32, 1999.

2. Schueremans L., “Assessment of existing structures using Adaptive Response SurfaceSampling - Theory and Applications”, Stichting Postacademisch Onderwijs, Probabilistischontwerpen, TUDelft, 5,6, 20 en 21 June, case 4, pp. 1-22, 2001.

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Acknowledgment

The research sponsorship from the Institute for the encouragement of Innovation by Science andTechnology in Vlaanderen (IWT-Vlaanderen: Instituut voor de aanmoediging van Innovatie doorWetenschap en Technologie in Vlaanderen) is gratefully acknowledged.